The number of sumsets in a finite field
|
|
- Posy Warner
- 5 years ago
- Views:
Transcription
1 The number of sumsets in a finite field Noga Alon Andrew Granville Adrián Ubis Abstract We prove that there are 2 p/2+op) distinct sumsets A + B in F p where A, B as p 1 Introduction For any subsets A and B of a group G we define the sumset A + B := {a + b : a A, b B} There are 2 n subsets of an n element additive group G and every one of them is a sumset, since A = A+{0} for every A G However if we restrict our summands to be slightly larger, then something surprising happens when G = F p : there are far fewer sumsets: Theorem 1 Let ψx) be any function for which ψx) and ψx) x/4 as x There are exactly 2 p/2+op) distinct sumsets in F p with summands of size ψp); that is, exactly 2 p/2+op) distinct sets of the form A + B with A, B ψp) where A, B F p Green and Ruzsa [GrRu] proved that there are only 2 p/3+op) distinct sumsets A + A in F p The count in Theorem 1 cannot be decreased by restricting the size of one of the sets: Theorem 2 For any given prime p and integer k satisfying k = op), there exists A F p with A = k for which there are at least 2 p/2+op) distinct sumsets of the form A + B with B F p These results do not give a good idea of the number of distinct sumsets of the form A + B, as B varies through the subsets of F p when A has a given small size Tel Aviv University, Tel Aviv 69978, Israel and IAS, Princeton, NJ, 08540, USA Research supported by the Israel Science Foundation, by a USA-Israel BSF grant, and by the Ambrose Monell Foundation nogaa@tauacil Université de Montréal, Montréal QC H3C 3J7, Canada L auteur est partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada andrew@dmsumontrealca Universidad de La Rioja, Logroño 26004, Spain Supported by a Spanish MEC grant and by a Comunidad de Madrid-Univ Autónoma Madrid grant adrianubis@gmailcom 1
2 Theorem 3 For each fixed integer k 1 there exists a constant µ k [ 2, 2] such that max A F p, A =k #{A + B : B F p} = µ p+op) k 1) We have µ 1 = 2, µ 2 := , the real root of x 3 2x 2 + x 1 and, for each fixed integer k 3, we have µ k k ) log k 2 + O 2) k Moreover µ k 5 5 / ) 1/5 = for all k 2, so that if A 2 then #{A + B : B F p } p+op) Remark: With a more involved method the constant in the last bound can be improved to see [Ubi]) We immediately deduce the following complement to Theorem 1: Corollary 1 Fix integer k 1 Let µ k = max l k µ l There are exactly µ k )p+op) distinct sumsets in F p with summands of size k The existence of µ k is deduced from the following result involving sumsets over the integers Define SA, G) to be the number of distinct sumsets A+B with B G; above we have looked at SA, F p ), but now we look at SA, {1, 2,, N}): Proposition 1 For any finite set of non-negative integers A with largest element L, there exists a constant µ A such that SA, {1, 2,, N}) = µ N+OL) A Moreover µ k = sup A Z 0 A =k By Theorem 3 or by Theorems 1 and 2 taken together) we know that µ k 2 as k In fact we believe that it does so monotonically: µ A Conjecture 1 We have µ 1 = 2 > µ 2 > µ 3 > > µ k > > 2 If this is true then µ k = µ k, evidently One can ask even more precise questions, for example for the number of distinct sumsets A + B where the sizes of A and B are given: Define S k,l G) = #{A + B : A, B G, A = k, B = l} for any integers k, l > 1 By Theorem 1 we know that if k, l as p then S k,l F p ) 2 p/2+op) We wish to determine for which values of k and l we have that S k,l F p ) 2 p/2+op) The Cauchy-Davenport Theorem [Cau] says that for any A, B F p we have A + B minp, A + B 1), hence S k,l F p ) = O1) whenever k + l > p O1) Let us see what we can say otherwise 2
3 Theorem 4 Let φ = and let ψx) be any function for which ψx) as x i) If k + l p then S k,l F p ) [p/2] k+l 2) / min{k, l} If k + l p/2φ then S k,l F p ) p O1) ) [p/2] k+l If φp/3 + O1) > k + l > p/2φ then S k,l F p ) φ p k l /p If p k + l φp/3 + O1) then S k,l F p ) pφ p k l /p + 1 k l) In summary, if k + l p then S k,l F p ) p O1) max [p h)/2] ) h k+l h ii) For any integers with k, l ψp) and p k l p, we have S k,l F p ) x x such that 2 p x x k+l) In particular, if k, l ψp) then if and only if k + l p/4 k+l) 1+o1) with S k,l F p ) = 2 p/2+op) 3) Note that Theorem 4ii) cannot hold for k + l very close to p by the last estimate in Theorem 4i) The structure of sumsets has a rich history, from Cauchy [Cau] onwards, and has been studied from several different perspectives Most important are lower bounds on the size of the sumset, the lattice structure of A and B when the size of the sumset A + B is not much larger than that of A and B ie the Freiman-Ruzsa theorem), and the discovery of long arithmetic progressions in the sumset A + B when it is fairly small From our problem, many questions naturally arise: Give a precise asymptotic for the number of sumsets in F p as well as for the number of sumsets A + A in F p Which sets S have at least 2 cp representations as A + B for a given c > 0, and in particular for c = 1/2? Can one quickly identify a sumset S in F p, where S = A + B with A, B k? Perhaps though this seems unlikely) any sumset contains enough structure that is quickly identifiable? Perhaps most non-sumsets are easily identifiable in that they lack certain structure? We do know [Al] that any complement of a set of size c p log p is a sumset A + A for some A F p, for some absolute constant c > 0 Given a set S for which there exist sets A, B with A = k, B = l such that A+B = S, can one find such a pair A, B quickly? Can one quickly identify those sumsets in F p which have many representations as A+B? 3
4 Estimate the size of the smallest possible collection C k of sets in F p such that if S = A+B where A, B k then there exist A, B C k for which S = A + B Perhaps even something stronger than Conjecture 1 is true: for any A Z of size 1 < k <, does there exist a A such that µ A\{a} > µ A? 2 Lower bounds For a given integer k let A = {0, [p k)/2] + 1, [p k)/2] + 2,, [p k)/2] + k 1} For any subset B of {0, 1, 2,, [p k)/2]}, we see that A + B [0, p 1] and B = A + B) {0, 1, 2, [p k)/2]}, and thus the sets A + B are all distinct Hence there are at least 2 [p k)/2]+1 2 p k)/2 distinct sets A + B as B varies over the subsets of F p This implies Theorem 2, hence the lower bound in Theorem 1 when ψp) = op), and it also implies the lower bound µ k 2 in Theorem 3 Let A = {0} [x + 1,, x + k u 1] x + k u 1 + A 1 ) and B = B 1 [x + 1,, x + l v 1] {x + l v 1 + y} where A 1 [1, y] with A 1 = u, and B 1 [1, x] with B 1 = v, where u < k, y, and v < l, x Therefore B 1 = A+B) [1, x] and N+A 1 = A+B) N+[1, y]) for N = 2x+y+k+l u v 2 Also A + B [0, p 1] provided 2x + 2y + k + l u v 2 < p Therefore S k,l y x u) v) If k+l p/2φ then we select u = k 1, v = l 1, y = [pk 1)/2k+l 2)], x = p 1)/2 y This gives S k,l p O1) ) [p/2] k+l by Stirling s formula; Sk,l [p/2] k+l 2) / min{k, l} if k + l p If k + l > p/2φ then we select u = [kp + 1 k l)/ 5k + l)], v = [lp + 1 k l)/ 5k + l)], y = [φu], x = [φv] to obtain S k,l φ p k l /p + 1 k l) by Stirling s formula If k + l φp/3 + O1) then we change the above construction slightly: If instead we take B 1 [0, x 1] then there is a unique block of k + l u v 3 consecutive integers in A + B starting with 2x + 2 Now we can also consider the sums r + A) + B, for any r mod p); notice that we can identify the value of r from A + B, since the longest block of consecutive integers in A + B starts with 2x r Hence S k,l pφ p k l /p + 1 k l) These last three paragraphs together imply the first part of Theorem 4 Now, given k p/4, select l = [p/4] k so that, by the above, there are p O1) [p/2] [p/4]) = 2 p/2 p O1) distinct sumsets A + B as A and B vary over the subsets of F p of size k and l respectively This implies the lower bound in Theorem 1 4
5 3 First upper bounds In this section we shall use a combinatorial argument to bound the number of sumsets A + B whenever A is small, in which case we can consider A fixed Throughout we let r C+A n) and r C A n)) denote the number of representations of n as c + a respectively, c a) with a A and c C Proposition 2 Let G be an abelian group of order n and let A G be a subset of size k 2 Then n ) n #{A + B : B G} n min min{2 n j, 2 [jk/k l+1)] } 4) 2 l k [j/l] j=0 Proof Given a set B we order the elements of B by greed, selecting any b 1 B, and then b 2 B so as to maximize A + {b 2 }) \ A + {b 1 }), then b 3 B so as to maximize A + {b 3 }) \ A + {b 1, b 2 }), etc Let B l be the set of b i such that A + {b 1, b 2,, b i } contains at least l more elements than A + {b 1, b 2,, b i 1 }, and suppose that B l + A = j By definition j = B l +A l B l, so that B l [j/l] and so there are no more than n i [j/l] i) choices for ) n n [j/l]) Next we have to determine B l Note that j/l n/2, for l 2, and so n i [j/l] i the number of possibilities for A + B given B l and hence B l + A): Our first argument: Since B l + A B + A G, the number of such sets A + B is at most the total number of sets H for which B l + A H G, which equals 2 n j Our second argument: Let C = B l + A, and let D be the set of d G for which r C A d) k + 1 l If b B \ B l then r C A b) = b + A) B l + A) k + 1 l, so that b D Hence B \ B l ) D, and so there are 2 D possible sets B \ B l, and hence B, and hence A + B Now D k + 1 l) r C A d) = A C = kj, d G so that D kj/k + 1 l), and the result follows Simplifying the upper bound: The upper bound in Proposition 2 is evidently ) n n 2 min max min{2 n j, 2 [jk/k l+1)] } 2 l k 0 j n [j/l] Now n [j/l]) 2 [jk/k l+1)] is a non-decreasing function of j, as l 2, and so the above is ) n n 2 min 2 n j 2 l k [j/l] max k l+1) 2k l+1) n j n The j + l)th term equals the jth term times n [j/l])/2 l [j/l] + 1) This is < 1 if and only if n < 2 l + 1)[j/l] + 2 l Now 2 l + 1)[j/l] + 2 l > 2l + 1) j 2l + 1) k l + 1) l l 2k l + 1) n, 5
6 and this is n unless l = k 4 Hence one minimizes by taking j = k l+1) n+o1) at a cost 2k l+1) of a factor of at most n Therefore our bound becomes n O1) νk n where ν k := min 2 l k ν k,l and ν k,l := 2 k l2k l + 1)) 2k l+1 k l + 1) k l+1 k l+1 2k l+1 l l2k l + 1) k l + 1)) l ) 1 2k l+1, using Stirling s formula A brief Maple calculation yields that ν k > 2 for all k 7 and ν 8 = , ν 9 = , ν 10 = ,, with ν k < 191 for k 12, and ν k decreasing rapidly and monotonically eg ν k < 19 for k 13, ν k < 18 for k 23, ν k < 17 for k 45, and ν k < 16 for k 117) In general taking l so that l 2 k log k/ log 2, one gets that ν k = 1 ) ) log 2 log k 2 exp 2 + o1), k which implies the upper bound in 2) of Theorem 3, as well as the upper bound implicit in Theorem 1 when min{ A, B } = op) 4 Upper bounds on S k,l F p ) using combinatorics The value of x in Theorem 4ii) must always lie in the range [p/2, p] since x k+l) 2 x Therefore if k + l = op) then the number of sumsets A + B is smaller than the number of possibilities for A and B so that S k,l F p ) p k ) ) p = l ) ) ) 1+o1) p x x 2 Ok+l) = 2 Ok+l) = k + l k + l k + l The Cauchy-Davenport Theorem states that A + B min{ A + B 1, p}, so that p ) ) p p S k,l F p ) j k + l 1 j=k+l 1 for k+l > 1/2+ɛ)p For the last part of Theorem 4, note that this is < 2 8p/17 for k+l 9p/10 Now we consider the case l, p k l p with k < op) and k For each fixed A of cardinality k and B of cardinality l, we proceed as in Proposition 2 taking l there as m here, and choosing m = ok) with m ): Hence there exists a subset B m B with A + B m = j and B m j/m p/m, and a subset D, determined by A and B m, with D kj j1 + Om/k)) and B \ B k+1 m m D Now A + B = A + B m ) A + B \ B m )) so the number of possibilities for A + B) \ A + B m ) is bounded above by the number of subsets of F p \ A + B m ), which is 2 p j, and also by the number of subsets of D with cardinality in the range [l [j/m], l], which is l ) ) D j 2 op), i l + k i=l [j/m] 6
7 since D j + op) and i = l + k + op) Hence the number of possible sumsets A + B is bounded by p k) 2 op), the number of possibilities for A, times p i [p/m] i) 2 op), the number of possibilities for B m, times 2 op) min{ j l+k), 2 p j }, the number of possibilities for A + B) \ A + B m ) This gives us the upper bound ) j S k,l F p ) 2 op) min{, 2 p j } = 2 1+o1))p x) 5) l + k where x is chosen as in Theorem 4ii), noting that p x p as l + k p 5 Sumsets from big sets We modify, simplify and generalize Green and Ruzsa s argument [GrRu], which they used to bound the number of sumsets A + A in F p : For a given set S, define ds := {ds : s S} Let G = Z/mZ For any A G define Âx) = a A eax/m) For a given positive integer L < m let H be the set of integers in the interval [ L 1), L 1] For a given integer d with 1 < dl < m we partition the integers in [1, m] as best as we can into arithmetic progressions with difference d and length L That is for 1 i d we have the progressions I i,k := {i + jd : kl j min{k + 1)L 1, [m i)/d]}} for 0 k [m i)/ld] We then let A L,d be the union of the I i,k that contain an element of A so that A A L,d ) Note that there are [m/l] + d such intervals I i,k Our goal is to prove the following analogy to Proposition 3 in [GrRu]: Proposition 3 If A, B Z/mZ, with α = A /m and β = B /m and m > 4L) 1+16αβL4 ɛ 2 2 ɛ 1 3, with L 3, 6) then there exists an integer d, with 1 d m/4l, such that A + B contains all those values of n for which r AL,d +B L,d n) > ɛ 2 m, with no more than ɛ 3 m exceptions In this paragraph we follow the proof of Proposition 3 in [GrRu] with the obvious modifications): Lemma 1 If A Z/mZ then there exists 1 d m/4l such that for all x Z/mZ Âx) 2 1 Ĥdx) 2L 1 ) 2 2 log 4L logm/4l) A m, with L 3, 7
8 Proof Sketch) Fix δ so that the right side above equals δm) 2, and hence δ 2/m Let R be the set of r Z/mZ such that Âr) δm; the result follows immediately for any x R By Parseval s inequality we have R δm) 2 r R Âr) 2 r Âr) 2 = m A, so that R δ 2 A /m Moreover, by the arithmetic-geometric mean inequality, we have ) R ) R 1 1 ) R m A Âr) 2 R Âr) 2 R Âr) 2 = R r R r R Consider the vectors v i [0, 1) R with rth coordinate ri/m mod 1) for each r R If we partition the unit interval for the rth coordinate into intervals of roughly equal length, all δm) 1/2 /4L 1) Âr) 1/2 which is 1/4L 1)), then, by the pigeonhole principle, two such vectors, with 0 i < j m/4l, lie in the same intervals since 1 + 4L 1) r R = Âr) δm 4L 1/2 4L r R Âr) δm r 1/2 ) ) 1/4 R A /m 4L) A /δ2m = m δ 2 R 4L, using the last displayed equation Therefore for d = j i we have ) 1/2 rd m 1 δm 4L 1 Âr) 4L ) R ) R /4 m A δm) 1/2 R for all r R, where t is the shortest distance from t to an integer Now Re1 et)) 2π 2 t 2 and jt j t, so that 1 Ĥdx) 2L 1 = 1 2L 1 2π2 2L 1 L 1 j= L 1) L 1 j= L 1) )) jdx 1 e m 2 jdx m 2π2 L 2 dx 3 m If x R then, by combining the last two displayed equations, this is 2π2 L 2 3 The result follows since 1 + Ĥdr) 2L 1 1 4L 1) 2 δm Âx) 2 as Ĥdx) 2L 1 8 δm 2 Âx) 2
9 Proof of Proposition 3 By Parseval s formula, and then Lemma 1 we have r A+Bn) r A+dH+B+dHn) 2L 1) 2 n 2 = 1 m Âx) 2 ˆBx) 2 1 x Ĥdx) 2L 1 ) 2 2 log 4L logm/4l) A x ˆBx) 2 = log 4L logm/4l) A B m ɛ2 2ɛ 3 m 3 16L 4 in this range for m Here r A+dH+B+dH n) denotes the number of representations of n as a + di + b + dj with a A, b B and i, j H) Now if g A L,d then there exists j H such that g + dj A, by definition, and hence r A+dH g) 1 Therefore r A+dH g) r AL,d g) for all g G, so that r A+dH+B+dH n) r AL,d +B L,d n) for all n Therefore if N is the set of n A + B such that r AL,d +B L,d n) > ɛ 2 m, then r A+dH+B+dH n) > ɛ 2 m and the above yields ɛ 2 m) 2 N 2L 1) ɛ2 2ɛ 3 m L 4 so that N ɛ 3 m Next we prove a combinatorial lemma based on Proposition 5 of [GrRu]: Proposition 4 For any subsets C, D of F p, and any m r min C, D ), there are at least min C + D, p) r m 1)p/r values of n mod p) such that r C+D n) m Proof Pollard s generalization of the Cauchy-Davenport Theorem [Pol] states that min{r, r C+D n)} r minp, C + D r) r[minp, C + D ) r] n The left hand side here is m 1)p N m ) + rn m where N m is the number of n mod p) such that r C+D n) m The result follows since p N m p Proof of upper bounds on S k,l F p ) using Fourier analysis: Suppose that L is given and d p/4l, and that M and N are unions of some of the arithmetic progressions I i,j Note that there are 2 p/l+d such sets M given d), and hence a total of e Op/L) possibilities for d, M and N We now bound the number of distinct sumsets A + B for which A L,d = M and B L,d = N in two different ways: First, since A M and B N there can be no more than ) M N ) k l M + N ) k+l 2 M + N such pairs 9
10 Second, select 2ɛ 1 p min M, N ) and 2ɛ 3 p max M, N ) Let Q be the values of n mod p) such that r M+N n) ɛ 2 1p Taking r = ɛ 1 p and m = ɛ 2 1p in Proposition 4, we have Q R := min M + N, p) 2ɛ 1 p By Proposition 3, A + B is given by Q less at most ɛ 3 p elements, union some subset of F p \ Q Hence the number of distinct sumsets A + B is [ɛ 3 p] ) Q 2 p Q i i=0 ) ) Q p2 p Q p2 maxp M N,0)+2ɛ 1p p [ɛ 3 p] [ɛ 3 p] as Q R > 2ɛ 3 p If M + N p/2 then the number of sumsets is 2 p/2 by the first argument Let L = [log p) 1/10 ] and ɛ 1 = ɛ 3 = 1/2L If A, B > p/l then M A > 2ɛ 1 p and N B > 2ɛ 1 p, so the second argument is applicable; therefore if M + N > p/2 then the number of sumsets is 2 p/2 L Op/L) Hence the total number of sumsets A + B with A, B > p/l is at most 2 p/2 L Op/L) which implies the upper bound in Theorem 1 taken together with the argument, for min{ A, B } = op), given at the end of section 3) Assume that l k p/log p) 1/4 with p k l p We select ɛ 1 = k/2p log log p, ɛ 3 = l/2p log log p and L = [log p) 1/20 ], so that the second argument above is applicable Taking x = M + N we have that { ) } x S k,l F p ) max min, 2 maxp x,0) 1/ɛ 3 ) Oɛ3p) = 2 1+o1))p x) 2 op) 0 x 2p k + l as in 5) This completes the proof of Theorem 4ii), combined with the results of the previous section Finally, 3) follows noting that x p/2 unless k + l p/4, in which case x p/2 6 Sumsets in finite fields and the integers Let A F p be of given size k 2, and let d = [p 1 1/k ] Consider the sets ia, the least residues of ia, a A, for 0 i p 1 Two, say ia and ja with i j mod p), must have those least residues between the same two multiples of p 1 1/k for each a A since there are < p/p 1 1/k ) k = p possibilities), and so the least residues of la, a A, with l = i j are all d in absolute value Hence the elements of d + la are all integers in [0, 2d]; and SA, F p ) = Sd + la, F p ) as may be seen by mapping A + B d + la) + lb) Hence we may assume, without loss of generality, that A is a set of integers in [0, L] where L 2p 1 1/k The case k = 2 is of particular interest since then SA, F p ) = S{0, 1}, F p ) by taking l = 1/b a), d = al when A = {a, b} We now compare SA, F p ) with SA, {1, 2,, p}) When we reduce A + B, where A {0,, L} and B {0,, p 1} are sets of integers, modulo p, the reduction only affects the residues in {0,, L 1} mod p) Hence SA, {1, 2,, p})2 L SA, F p ) SA, {1, 2,, p}) 7) 10
11 Now suppose A {0,, L} is a set of integers Suppose that Mr N < Mr + 1) for positive integers M, r, N We see that SA, {1, 2,, N}) SA, {1, 2,, Mr + 1)}) SA, {1, 2,, M}) r+1, the last inequality coming since the sumsets A + B with B {1, 2,, Mr + 1)} are the union of the sumsets A + B i with B i {Mi + 1, 2,, Mi + 1)} for i = 0, 1, 2,, r In particular for m A N) := SA, {1, 2,, N}) 1/N we have m A N) m A M) 1+1/r This implies that lim sup N m A N) m A M) for any fixed M, and then lim sup N m A N) = lim inf N m A N) so the limit, say µ A, exists and satisfies SA, {1, 2,, M}) µ M A 8) In the other direction we note that if B = i B i where B i {M + L)i + 1, M + L)i + 2,, M + L)i + M} then distinct {A + B i } 0 i r 1 give rise to distinct A + B Hence SA, {1, 2,, M}) r SA, {1, 2,, rm + L)}) and letting r we have Finally, by the inequalities 7), 8) and 9) we arrive at SA, {1, 2,, M}) µ M+L A 9) SA, F p ) = µ p A eol) = µ p A eop1 1/k) = µ p+op) A This proves Proposition 1, as well as the first part of Theorem 3 61 Precise bounds when k = 2 By the previous section we know that µ 2 = µ {0,1} Now S is a sumset of the form {0, 1} + B if and only if, when one writes the sequence of 0 s and 1 s given by s n = 1 if n S, otherwise s n = 0 if n S, there are no isolated 1 s Let C n be the number of sequences of 0 s and 1 s of length n such that there are no isolated 1 s, so that S{0, 1}, {1, 2,, N}) = C N+1 We can determine C n+1 by induction: If the n + 1)th element added is a 0 then it can be added to any element of C n If the n + 1)th element added is a 1 then the nth digit must be a 1, and then we either have an element of C n 1 or the next two digits are 1 and 0 followed by any element of C n 3 Hence C n+1 = C n + C n 1 + C n 3 with C 1 = 1, C 2 = 2, C 3 = 4, C 4 = 7 In fact it is easily checked, by induction, that C n+1 = 2C n C n 1 + C n 2 which is explained by the fact that x 4 x 3 x 2 1 = x+1)x 3 2x 2 +x 1), where the higher degree polynomials are characteristic polynomials for the recurrence sequence), and hence C n cµ n 2 for some constant c > 0, with µ 2 as in Theorem 3, implying a strong form of the first part of Theorem 3 for k = 2 By a more precise analysis we could even estimate the number of sets C = {0, 1} + B with either C or B of given size 11
12 62 Precise bounds when k = 3 It is not hard to generalize the procedure for the case {0, 1} to any A Z finite, namely to prove that µ A is the root of a polynomial with integer coefficients and degree smaller than 2 2L+1 when A {0, 1,, L}) In the special case of three elements is enough to deal with A = {0, a, b} for a, b coprime positive integers We can show that µ {0,a,b} µ as a + b, where we define µ = lim p #{B + {0, 0), 1, 0), 0, 1)} : B F p F p } 1/p2, which one can prove exists, and is < µ 2 Therefore either µ 3 = µ {0,a,b} for some a and b or µ 3 = µ Maple experimentation leads us to guess that µ 3 = µ {0,1,4} = 16863, a root of an irreducible polynomial of degree 21 All this is detailed in Chapter 3 of the third author s PhD thesis [Ubi] 63 Lower bounds on µ k That µ k 2 follows by choosing A = 1 2A with A Z any finite set Let A k = {1, 3,, 3 k 1 }, and write B {1, 2,, 3n} as B = 3B 0 3B 1 1) with B 0, B 1 {1, 2,, n} Since A k+1 = 1 3A k we have B + A k+1 ) \ 3Z = 3B 1 + A k ) 1) 3B 0 + 1), which shows that SA k+1, {1, 2,, 3n}) SA k, {1, 2,, n}) 2 n, and so µ Ak µ 1 3 A k Since µ A1 = 2, an induction argument implies µ k µ Ak 2 1/2+31 k /2, which gives the lower bound for µ k in 2) 7 A non-trivial bound for fixed k 2 Let A be any set of given size k 2 in F p For any two distinct elements a, b A we can map x x a)/b a) so that 0, 1 A, and this will not effect the count of the number of sumsets containing A The number of sumsets C = A + B with B F p is obviously bounded above by { # B : B 2p } { + # C : C 3p } 5 5 { +# C : B : 2p 5 < B < C < 3p 5 } and B + {0, 1} C 12
13 The first two terms have size 2p p [2p/5]), the third requires some work: We observe that such C must have at least 2p/5 pairs of consecutive elements; so if c is the smallest integer 1 that belongs to C then we suppose that C = m k=1 c + I k) and C = m k=1 c + J k) where I 1, J 1, I 2, J 2,, I m, J m is a partition of {0,, p 1} into non-empty set of integers from intervals taken in order Any such set partition will do provided, for i k = I k and j k = J k, we have each i k, j k 1, m 3p 5 i k m + 2p 5, k=1 since C = m k=1 i k and m k=1 i k 1) B, and m k=1 i k + m k=1 j k = p Now there are p possible values for c, and the number of possible sets of values of i k such that m ) k=1 i k = x is, and of jk is ) p x 1 Therefore the number of possible such C is x 1 m 1 m 1 p 2 p m p/5 m p/5 2p/5+m x 3p/5 x 1 m 1 ) ) p 2 p 2 p 3 2m 2 [2p/5 2] ) p x 1 m 1 ) ) p p 3 [2p/5] Note that a c ) b) d a+c b+d) follows from defining a b) to be the number of ways of choosing b elements from a) Hence the number of sumsets A + B is p 3 p [2p/5]) = c p+op) where c = 5 5 / ) 1/5 = This implies the bound µ k c for all k 2 of Theorem 3; and we deduce the last part of Theorem 3 immediately from this taken together with Theorem 1 References [Al] N Alon Large sets in finite fields are sumsets J Number Theory, 126, , 2007 [Cau] A Cauchy Recherches sur les nombres J École Polytech, , 1813 [GrRu] B Green, I Z Ruzsa Counting sumsets and sum-free sets modulo a prime Studia Sci Math Hungar, , 2004 [Pol] J M Pollard A generalisation of the theorem of Cauchy and Davenport J London Math Soc 2), , 1974 [Ubi] A Ubis Cuestiones de la Aritmética y del Análisis Armónico PhD thesis, Universidad Autónoma de Madrid, Madrid, 2006 English translation available at the web address 13
GENERALIZATIONS OF SOME ZERO-SUM THEOREMS. Sukumar Das Adhikari Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad , INDIA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A52 GENERALIZATIONS OF SOME ZERO-SUM THEOREMS Sukumar Das Adhikari Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad
More informationSome zero-sum constants with weights
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 118, No. 2, May 2008, pp. 183 188. Printed in India Some zero-sum constants with weights S D ADHIKARI 1, R BALASUBRAMANIAN 2, F PAPPALARDI 3 andprath 2 1 Harish-Chandra
More informationSolving a linear equation in a set of integers II
ACTA ARITHMETICA LXXII.4 (1995) Solving a linear equation in a set of integers II by Imre Z. Ruzsa (Budapest) 1. Introduction. We continue the study of linear equations started in Part I of this paper.
More informationA proof of Freiman s Theorem, continued. An analogue of Freiman s Theorem in a bounded torsion group
A proof of Freiman s Theorem, continued Brad Hannigan-Daley University of Waterloo Freiman s Theorem Recall that a d-dimensional generalized arithmetic progression (GAP) in an abelian group G is a subset
More informationSUMSETS MOD p ØYSTEIN J. RØDSETH
SUMSETS MOD p ØYSTEIN J. RØDSETH Abstract. In this paper we present some basic addition theorems modulo a prime p and look at various proof techniques. We open with the Cauchy-Davenport theorem and a proof
More informationCLASSIFICATION THEOREMS FOR SUMSETS MODULO A PRIME. 1. Introduction.
CLASSIFICATION THEOREMS FOR SUMSETS MODULO A PRIME HOI H. NGUYEN AND VAN H. VU Abstract. Let Z p be the finite field of prime order p and A be a subsequence of Z p. We prove several classification results
More informationSzemerédi-Trotter theorem and applications
Szemerédi-Trotter theorem and applications M. Rudnev December 6, 2004 The theorem Abstract These notes cover the material of two Applied post-graduate lectures in Bristol, 2004. Szemerédi-Trotter theorem
More informationResearch Problems in Arithmetic Combinatorics
Research Problems in Arithmetic Combinatorics Ernie Croot July 13, 2006 1. (related to a quesiton of J. Bourgain) Classify all polynomials f(x, y) Z[x, y] which have the following property: There exists
More informationANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99
ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS N. HEGYVÁRI, F. HENNECART AND A. PLAGNE Abstract. We study the gaps in the sequence of sums of h pairwise distinct elements
More informationWalk through Combinatorics: Sumset inequalities.
Walk through Combinatorics: Sumset inequalities (Version 2b: revised 29 May 2018) The aim of additive combinatorics If A and B are two non-empty sets of numbers, their sumset is the set A+B def = {a+b
More informationA lattice point problem and additive number theory
A lattice point problem and additive number theory Noga Alon and Moshe Dubiner Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract
More informationSETS WITH MORE SUMS THAN DIFFERENCES. Melvyn B. Nathanson 1 Lehman College (CUNY), Bronx, New York
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A05 SETS WITH MORE SUMS THAN DIFFERENCES Melvyn B. Nathanson 1 Lehman College (CUNY), Bronx, New York 10468 melvyn.nathanson@lehman.cuny.edu
More informationSTABLE KNESER HYPERGRAPHS AND IDEALS IN N WITH THE NIKODÝM PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 STABLE KNESER HYPERGRAPHS AND IDEALS IN N WITH THE NIKODÝM PROPERTY NOGA ALON, LECH DREWNOWSKI,
More informationSubset sums modulo a prime
ACTA ARITHMETICA 131.4 (2008) Subset sums modulo a prime by Hoi H. Nguyen, Endre Szemerédi and Van H. Vu (Piscataway, NJ) 1. Introduction. Let G be an additive group and A be a subset of G. We denote by
More informationTHE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Z p. Mario Huicochea CINNMA, Querétaro, México
#A8 INTEGERS 15A (2015) THE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Z p Mario Huicochea CINNMA, Querétaro, México dym@cimat.mx Amanda Montejano UNAM Facultad de Ciencias
More informationA misère-play -operator
A misère-play -operator Matthieu Dufour Silvia Heubach Urban Larsson arxiv:1608.06996v1 [math.co] 25 Aug 2016 July 31, 2018 Abstract We study the -operator (Larsson et al, 2011) of impartial vector subtraction
More informationWeighted Sequences in Finite Cyclic Groups
Weighted Sequences in Finite Cyclic Groups David J. Grynkiewicz and Jujuan Zhuang December 11, 008 Abstract Let p > 7 be a prime, let G = Z/pZ, and let S 1 = p gi and S = p hi be two sequences with terms
More informationMaximum union-free subfamilies
Maximum union-free subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called
More informationFermat numbers and integers of the form a k + a l + p α
ACTA ARITHMETICA * (200*) Fermat numbers and integers of the form a k + a l + p α by Yong-Gao Chen (Nanjing), Rui Feng (Nanjing) and Nicolas Templier (Montpellier) 1. Introduction. In 1849, A. de Polignac
More informationA LOWER BOUND FOR THE SIZE OF A MINKOWSKI SUM OF DILATES. 1. Introduction
A LOWER BOUND FOR THE SIZE OF A MINKOWSKI SUM OF DILATES Y. O. HAMIDOUNE AND J. RUÉ Abstract. Let A be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained
More informationA talk given at the University of California at Irvine on Jan. 19, 2006.
A talk given at the University of California at Irvine on Jan. 19, 2006. A SURVEY OF ZERO-SUM PROBLEMS ON ABELIAN GROUPS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093 People s
More informationThe Cauchy-Davenport Theorem for Finite Groups
arxiv:1202.1816v1 [math.co] 8 Feb 2012 The Cauchy-Davenport Theorem for Finite Groups Jeffrey Paul Wheeler February 2006 Abstract The Cauchy-Davenport theorem states that for any two nonempty subsetsaandb
More informationEquidivisible consecutive integers
& Equidivisible consecutive integers Ivo Düntsch Department of Computer Science Brock University St Catherines, Ontario, L2S 3A1, Canada duentsch@cosc.brocku.ca Roger B. Eggleton Department of Mathematics
More informationCOUNTING NUMERICAL SEMIGROUPS BY GENUS AND SOME CASES OF A QUESTION OF WILF
COUNTING NUMERICAL SEMIGROUPS BY GENUS AND SOME CASES OF A QUESTION OF WILF NATHAN KAPLAN Abstract. The genus of a numerical semigroup is the size of its complement. In this paper we will prove some results
More informationReconstructing integer sets from their representation functions
Reconstructing integer sets from their representation functions Vsevolod F. Lev Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel seva@math.haifa.ac.il Submitted: Oct 5, 2004;
More informationInduced subgraphs of prescribed size
Induced subgraphs of prescribed size Noga Alon Michael Krivelevich Benny Sudakov Abstract A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(g denote the maximum
More information64 P. ERDOS AND E. G. STRAUS It is possible that (1.1) has only a finite number of solutions for fixed t and variable k > 2, but we cannot prove this
On Products of Consecutive Integers P. ERDŐS HUNGARIAN ACADEMY OF SCIENCE BUDAPEST, HUNGARY E. G. STRAUSt UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA 1. Introduction We define A(n, k) = (n + k)!/n!
More informationProblems in additive number theory, V: Affinely inequivalent MSTD sets
W N North-Western European Journal of Mathematics E M J Problems in additive number theory, V: Affinely inequivalent MSTD sets Melvyn B Nathanson 1 Received: November 9, 2016/Accepted: June 26, 2017/Online:
More informationAdditive Combinatorics and Szemerédi s Regularity Lemma
Additive Combinatorics and Szemerédi s Regularity Lemma Vijay Keswani Anurag Sahay 20th April, 2015 Supervised by : Dr. Rajat Mittal 1 Contents 1 Introduction 3 2 Sum-set Estimates 4 2.1 Size of sumset
More informationAn upper bound for B 2 [g] sets
Journal of Number Theory 1 (007) 11 0 www.elsevier.com/locate/jnt An upper bound for B [g] sets Gang Yu Department of Mathematics, LeConte College, 153 Greene street, University of South Carolina, Columbia,
More informationRoth s Theorem on 3-term Arithmetic Progressions
Roth s Theorem on 3-term Arithmetic Progressions Mustazee Rahman 1 Introduction This article is a discussion about the proof of a classical theorem of Roth s regarding the existence of three term arithmetic
More informationON MATCHINGS IN GROUPS
ON MATCHINGS IN GROUPS JOZSEF LOSONCZY Abstract. A matching property conceived for lattices is examined in the context of an arbitrary abelian group. The Dyson e-transform and the Cauchy Davenport inequality
More informationCONSECUTIVE INTEGERS IN HIGH-MULTIPLICITY SUMSETS
CONSECUTIVE INTEGERS IN HIGH-MULTIPLICITY SUMSETS VSEVOLOD F. LEV Abstract. Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves
More informationMei-Chu Chang Department of Mathematics University of California Riverside, CA 92521
ERDŐS-SZEMERÉDI PROBLEM ON SUM SET AND PRODUCT SET Mei-Chu Chang Department of Mathematics University of California Riverside, CA 951 Summary. The basic theme of this paper is the fact that if A is a finite
More informationNORMAL NUMBERS AND UNIFORM DISTRIBUTION (WEEKS 1-3) OPEN PROBLEMS IN NUMBER THEORY SPRING 2018, TEL AVIV UNIVERSITY
ORMAL UMBERS AD UIFORM DISTRIBUTIO WEEKS -3 OPE PROBLEMS I UMBER THEORY SPRIG 28, TEL AVIV UIVERSITY Contents.. ormal numbers.2. Un-natural examples 2.3. ormality and uniform distribution 2.4. Weyl s criterion
More informationGROWTH IN GROUPS I: SUM-PRODUCT. 1. A first look at growth Throughout these notes A is a finite set in a ring R. For n Z + Define
GROWTH IN GROUPS I: SUM-PRODUCT NICK GILL 1. A first look at growth Throughout these notes A is a finite set in a ring R. For n Z + Define } A + {{ + A } = {a 1 + + a n a 1,..., a n A}; n A } {{ A} = {a
More informationThe 123 Theorem and its extensions
The 123 Theorem and its extensions Noga Alon and Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract It is shown
More informationCongruences involving product of intervals and sets with small multiplicative doubling modulo a prime
Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime J. Cilleruelo and M. Z. Garaev Abstract We obtain a sharp upper bound estimate of the form Hp o(1)
More informationEvgeniy V. Martyushev RESEARCH STATEMENT
Evgeniy V. Martyushev RESEARCH STATEMENT My research interests lie in the fields of topology of manifolds, algebraic topology, representation theory, and geometry. Specifically, my work explores various
More informationCourse MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography
Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups
More informationZero-sum square matrices
Zero-sum square matrices Paul Balister Yair Caro Cecil Rousseau Raphael Yuster Abstract Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the
More informationMATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.
MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationOn the number of cycles in a graph with restricted cycle lengths
On the number of cycles in a graph with restricted cycle lengths Dániel Gerbner, Balázs Keszegh, Cory Palmer, Balázs Patkós arxiv:1610.03476v1 [math.co] 11 Oct 2016 October 12, 2016 Abstract Let L be a
More informationOn the Infinitude of Twin-Primes Bao Qi Feng
On the Infinitude of Twin-Primes Bao Qi Feng Department of Mathematical Sciences, Kent State University at Tuscarawas 330 University Dr. NE, New Philadelphia, OH 44663, USA. Abstract In this article, we
More informationARITHMETIC PROGRESSIONS IN SPARSE SUMSETS. Dedicated to Ron Graham on the occasion of his 70 th birthday
ARITHMETIC PROGRESSIONS IN SPARSE SUMSETS Dedicated to Ron Graham on the occasion of his 70 th birthday Ernie Croot 1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 Imre Ruzsa
More informationarxiv: v2 [math.nt] 15 May 2013
INFINITE SIDON SEQUENCES JAVIER CILLERUELO arxiv:09036v [mathnt] 5 May 03 Abstract We present a method to construct dense infinite Sidon sequences based on the discrete logarithm We give an explicit construction
More informationTewodros Amdeberhan, Dante Manna and Victor H. Moll Department of Mathematics, Tulane University New Orleans, LA 70118
The -adic valuation of Stirling numbers Tewodros Amdeberhan, Dante Manna and Victor H. Moll Department of Mathematics, Tulane University New Orleans, LA 7011 Abstract We analyze properties of the -adic
More informationJeong-Hyun Kang Department of Mathematics, University of West Georgia, Carrollton, GA
#A33 INTEGERS 10 (2010), 379-392 DISTANCE GRAPHS FROM P -ADIC NORMS Jeong-Hyun Kang Department of Mathematics, University of West Georgia, Carrollton, GA 30118 jkang@westga.edu Hiren Maharaj Department
More informationLOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi
LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, China and Center for Combinatorics,
More informationWeek 15-16: Combinatorial Design
Week 15-16: Combinatorial Design May 8, 2017 A combinatorial design, or simply a design, is an arrangement of the objects of a set into subsets satisfying certain prescribed properties. The area of combinatorial
More informationNew upper bound for sums of dilates
New upper bound for sums of dilates Albert Bush Yi Zhao Department of Mathematics and Statistics Georgia State University Atlanta, GA 30303, USA albertbush@gmail.com yzhao6@gsu.edu Submitted: Nov 2, 2016;
More informationMinimal Paths and Cycles in Set Systems
Minimal Paths and Cycles in Set Systems Dhruv Mubayi Jacques Verstraëte July 9, 006 Abstract A minimal k-cycle is a family of sets A 0,..., A k 1 for which A i A j if and only if i = j or i and j are consecutive
More informationArithmetic Progressions with Constant Weight
Arithmetic Progressions with Constant Weight Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel e-mail: raphy@oranim.macam98.ac.il Abstract Let k n be two positive
More informationAdditive Latin Transversals
Additive Latin Transversals Noga Alon Abstract We prove that for every odd prime p, every k p and every two subsets A = {a 1,..., a k } and B = {b 1,..., b k } of cardinality k each of Z p, there is a
More information9 - The Combinatorial Nullstellensatz
9 - The Combinatorial Nullstellensatz Jacques Verstraëte jacques@ucsd.edu Hilbert s nullstellensatz says that if F is an algebraically closed field and f and g 1, g 2,..., g m are polynomials in F[x 1,
More informationCoins with arbitrary weights. Abstract. Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish to
Coins with arbitrary weights Noga Alon Dmitry N. Kozlov y Abstract Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish to decide if all the m given coins have the
More informationThe diameter of a random Cayley graph of Z q
The diameter of a random Cayley graph of Z q Gideon Amir Ori Gurel-Gurevich September 4, 009 Abstract Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators.
More informationNotes on Freiman s Theorem
Notes on Freiman s Theorem Jacques Verstraëte Department of Mathematics University of California, San Diego La Jolla, CA, 92093. E-mail: jacques@ucsd.edu. 1 Introduction Freiman s Theorem describes the
More informationHeights of characters and defect groups
[Page 1] Heights of characters and defect groups Alexander Moretó 1. Introduction An important result in ordinary character theory is the Ito-Michler theorem, which asserts that a prime p does not divide
More informationCourse 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography
Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups
More informationAndrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers
PRIMITIVE PRIME FACTORS IN SECOND-ORDER LINEAR RECURRENCE SEQUENCES Andrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers Abstract. For a class of Lucas sequences
More informationKarol Cwalina. A linear bound on the dimension in Green-Ruzsa s Theorem. Uniwersytet Warszawski. Praca semestralna nr 1 (semestr zimowy 2010/11)
Karol Cwalina Uniwersytet Warszawski A linear bound on the dimension in Green-Ruzsa s Theorem Praca semestralna nr 1 (semestr zimowy 2010/11) Opiekun pracy: Tomasz Schoen A LINEAR BOUND ON THE DIMENSION
More informationChapter 4. Measure Theory. 1. Measure Spaces
Chapter 4. Measure Theory 1. Measure Spaces Let X be a nonempty set. A collection S of subsets of X is said to be an algebra on X if S has the following properties: 1. X S; 2. if A S, then A c S; 3. if
More informationAN EASY GENERALIZATION OF EULER S THEOREM ON THE SERIES OF PRIME RECIPROCALS
AN EASY GENERALIZATION OF EULER S THEOREM ON THE SERIES OF PRIME RECIPROCALS PAUL POLLACK Abstract It is well-known that Euclid s argument can be adapted to prove the infinitude of primes of the form 4k
More informationSum of dilates in vector spaces
North-Western European Journal of Mathematics Sum of dilates in vector spaces Antal Balog 1 George Shakan 2 Received: September 30, 2014/Accepted: April 23, 2015/Online: June 12, 2015 Abstract Let d 2,
More informationSpanning and Independence Properties of Finite Frames
Chapter 1 Spanning and Independence Properties of Finite Frames Peter G. Casazza and Darrin Speegle Abstract The fundamental notion of frame theory is redundancy. It is this property which makes frames
More informationNon-averaging subsets and non-vanishing transversals
Non-averaging subsets and non-vanishing transversals Noga Alon Imre Z. Ruzsa Abstract It is shown that every set of n integers contains a subset of size Ω(n 1/6 ) in which no element is the average of
More informationAhlswede Khachatrian Theorems: Weighted, Infinite, and Hamming
Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of
More informationSome problems related to Singer sets
Some problems related to Singer sets Alex Chmelnitzki October 24, 2005 Preface The following article describes some of the work I did in the summer 2005 for Prof. Ben Green, Bristol University. The work
More informationThe Algorithmic Aspects of the Regularity Lemma
The Algorithmic Aspects of the Regularity Lemma N. Alon R. A. Duke H. Lefmann V. Rödl R. Yuster Abstract The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in
More informationOut-colourings of Digraphs
Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.
More informationArithmetic progressions in sumsets
ACTA ARITHMETICA LX.2 (1991) Arithmetic progressions in sumsets by Imre Z. Ruzsa* (Budapest) 1. Introduction. Let A, B [1, N] be sets of integers, A = B = cn. Bourgain [2] proved that A + B always contains
More informationarxiv:math/ v2 [math.nt] 3 Dec 2003
arxiv:math/0302091v2 [math.nt] 3 Dec 2003 Every function is the representation function of an additive basis for the integers Melvyn B. Nathanson Department of Mathematics Lehman College (CUNY) Bronx,
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationOn zero-sum partitions and anti-magic trees
Discrete Mathematics 09 (009) 010 014 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/disc On zero-sum partitions and anti-magic trees Gil Kaplan,
More informationSUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM MOD P
SUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM MOD P TODD COCHRANE AND CHRISTOPHER PINNER Abstract. Let γ(k, p) denote Waring s number (mod p) and δ(k, p) denote the ± Waring s number (mod p). We use
More informationJoshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA.
CONTINUED FRACTIONS WITH PARTIAL QUOTIENTS BOUNDED IN AVERAGE Joshua N. Cooper Department of Mathematics, Courant Institute of Mathematics, New York University, New York, NY 10012, USA cooper@cims.nyu.edu
More informationOn non-hamiltonian circulant digraphs of outdegree three
On non-hamiltonian circulant digraphs of outdegree three Stephen C. Locke DEPARTMENT OF MATHEMATICAL SCIENCES, FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FL 33431 Dave Witte DEPARTMENT OF MATHEMATICS, OKLAHOMA
More informationPRIMITIVE PRIME FACTORS IN SECOND-ORDER LINEAR RECURRENCE SEQUENCES
PRIMITIVE PRIME FACTORS IN SECOND-ORDER LINEAR RECURRENCE SEQUENCES Andrew Granville To Andrzej Schinzel on his 75th birthday, with thanks for the many inspiring papers Abstract. For a class of Lucas sequences
More informationChapter Four Gelfond s Solution of Hilbert s Seventh Problem (Revised January 2, 2011)
Chapter Four Gelfond s Solution of Hilbert s Seventh Problem (Revised January 2, 2011) Before we consider Gelfond s, and then Schneider s, complete solutions to Hilbert s seventh problem let s look back
More informationRestricted set addition in Abelian groups: results and conjectures
Journal de Théorie des Nombres de Bordeaux 17 (2005), 181 193 Restricted set addition in Abelian groups: results and conjectures par Vsevolod F. LEV Résumé. Nous présentons un ensemble de conjectures imbriquées
More informationThe Real Number System
MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely
More informationp-adic Analysis Compared to Real Lecture 1
p-adic Analysis Compared to Real Lecture 1 Felix Hensel, Waltraud Lederle, Simone Montemezzani October 12, 2011 1 Normed Fields & non-archimedean Norms Definition 1.1. A metric on a non-empty set X is
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationP -adic root separation for quadratic and cubic polynomials
P -adic root separation for quadratic and cubic polynomials Tomislav Pejković Abstract We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible
More informationA PROOF OF ROTH S THEOREM ON ARITHMETIC PROGRESSIONS
A PROOF OF ROTH S THEOREM ON ARITHMETIC PROGRESSIONS ERNIE CROOT AND OLOF SISASK Abstract. We present a proof of Roth s theorem that follows a slightly different structure to the usual proofs, in that
More informationPacking Ferrers Shapes
Packing Ferrers Shapes Noga Alon Miklós Bóna Joel Spencer February 22, 2002 Abstract Answering a question of Wilf, we show that if n is sufficiently large, then one cannot cover an n p(n) rectangle using
More informationPowers of 2 with five distinct summands
ACTA ARITHMETICA * (200*) Powers of 2 with five distinct summands by Vsevolod F. Lev (Haifa) 0. Summary. We show that every sufficiently large, finite set of positive integers of density larger than 1/3
More informationPair dominating graphs
Pair dominating graphs Paul Balister Béla Bollobás July 25, 2004 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152 USA Abstract An oriented graph dominates pairs if for every
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More informationSUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM OVER FINITE FIELDS
#A68 INTEGERS 11 (011) SUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM OVER FINITE FIELDS Todd Cochrane 1 Department of Mathematics, Kansas State University, Manhattan, Kansas cochrane@math.ksu.edu James
More informationA proof of a partition conjecture of Bateman and Erdős
proof of a partition conjecture of Bateman and Erdős Jason P. Bell Department of Mathematics University of California, San Diego La Jolla C, 92093-0112. US jbell@math.ucsd.edu 1 Proposed Running Head:
More informationarxiv:math/ v1 [math.nt] 9 Feb 1998
ECONOMICAL NUMBERS R.G.E. PINCH arxiv:math/9802046v1 [math.nt] 9 Feb 1998 Abstract. A number n is said to be economical if the prime power factorisation of n can be written with no more digits than n itself.
More information#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES
#A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of
More informationThe number of edge colorings with no monochromatic cliques
The number of edge colorings with no monochromatic cliques Noga Alon József Balogh Peter Keevash Benny Sudaov Abstract Let F n, r, ) denote the maximum possible number of distinct edge-colorings of a simple
More informationKatarzyna Mieczkowska
Katarzyna Mieczkowska Uniwersytet A. Mickiewicza w Poznaniu Erdős conjecture on matchings in hypergraphs Praca semestralna nr 1 (semestr letni 010/11 Opiekun pracy: Tomasz Łuczak ERDŐS CONJECTURE ON MATCHINGS
More informationOn a Balanced Property of Compositions
On a Balanced Property of Compositions Miklós Bóna Department of Mathematics University of Florida Gainesville FL 32611-8105 USA Submitted: October 2, 2006; Accepted: January 24, 2007; Published: March
More informationCovering Subsets of the Integers and a Result on Digits of Fibonacci Numbers
University of South Carolina Scholar Commons Theses and Dissertations 2017 Covering Subsets of the Integers and a Result on Digits of Fibonacci Numbers Wilson Andrew Harvey University of South Carolina
More informationOn a combinatorial method for counting smooth numbers in sets of integers
On a combinatorial method for counting smooth numbers in sets of integers Ernie Croot February 2, 2007 Abstract In this paper we develop a method for determining the number of integers without large prime
More informationOn the number of co-prime-free sets. Neil J. Calkin and Andrew Granville *
On the number of co-prime-free sets. by Neil J. Calkin and Andrew Granville * Abstract: For a variety of arithmetic properties P such as the one in the title) we investigate the number of subsets of the
More information