POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS

Transcription

1 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS ELAD AIGNER-HOREV AND HIỆP HÀN Abstract. We prove transference results for sparse random and pseudo-random subsets of Z N,whichareanalogoustothequantitativeversionofthewellknown Furstenberg-Sárközy theorem due to Balog, Pintz, Pelikán, and Szemerédi. In the dense case, Balog et al showed that there is a constant C>0such that for all integer k 2anysubsetofthefirstN integers of density at least C(log N) 4 log log log log N contains a configuration of the form {x, x + d k } for some integer d>0. Let [Z N ] p denote the random set obtained by choosing each element from Z N with probability p independently. Our first result shows that for p > N /k+o() asymptotically almost surely any subset A [Z N ] p (N prime) of density A /pn (log N) 5 log log log log N contains the polynomial configuration {x, x + d k },0<dappleN /k. This improves on a result of Nguyen in the setting of Z N. Moreover, let k 2beanintegerandlet > > 0berealnumbers satisfying +( )/(2 k+ 3) >. Let Z N (N prime) be a set of size at least N and linear bias at most N. Then our second result implies that every A with positive relative density contains the polynomial configuration {x, x + d k },0<dapple N /k. For instance, for squares, i.e., k = 2, andassumingthebestpossiblepseudo- randomness = /2 ourresultappliesassoonas > 0/.. Introduction A classical result in additive combinatorics, proved independently by Sárközy [20] and Furstenberg [5], states in its qualitatively form that subsets of the first N integers with positive density contains a pair which di er by a perfect k-th power, i.e., a pair {x, x+d k } for some d>0. Improving on [20, 5] it is shown by Pintz, Steiger, Szemerédi [8] and by Balog, Pintz, Pelikán, Szemerédi [] that this conclusion already holds for sets of much smaller density. Theorem. There exists a constant C>0such that for all integer k sets A {,...,N} with density A /N C(log N) 4 log log log log N there exist integers x, d > 0, such that {x, x + d k } A. 2 and all To appreciate the bound in Theorem we note it is conjectured the bound is not far from best possible and that the largest set without the configuration x, x + d k has size (N " ) for all " > 0. A construction due to Ruzsa [9] shows that this is The second author was supported by FAPESP (Proc. 200/6526-3).

2 2 ELAD AIGNER-HOREV AND HIỆP HÀN true for " > Moreover, it is of interest to improve upon the constant /4 in Theorem (see [0]). Of course the result also holds for the setting Z N instead of {,...,N} and in this note we prove results analogous to Theorem for Z N with prime N. More precisely, we will study the case when the dense host {,...,N} or Z N is replaced by sparse random subsets or by pseudo-random subsets of Z N defined via small linear bias. These types of results are commonly called transference in which extremal results known for dense hosts are transferred or carried over to sparse hosts taken from a well-behaved universe like random or pseudo-random subsets of the original dense host. For sparse random hosts transference has been studied extensively in the last decades and the recent breakthroughs [22, 3] (see also [2, 2]) led to a much better understanding of the subject. First to extend Theorem to random hosts were Hamel and Laba [4]. Their result was improved by Nguyen [7] later on. Let [Z N ] p denote the random set obtained by choosing each element from Z N with probability p independently. Nguyen [7] proved that asymptotically almost surely (i.e. with probability tending to one as N!) every relatively dense subset of [Z N ] p contains a configuration {x, x + d k } provided that p>cn /k for some constant C = C(k). Up to the multiplicative constant C the bound on p is best possible. Again, the result in [7] is stated for {,...,N} instead of Z N. Concerning transference to random host we show the following. Theorem 2. N is prime For all integer k 2 following holds asymptotically almost surely. Let p>n /k exp (log N) 9/0 and let A [Z N ] p, N prime, be such that A /pn (log N) 5 log log log log N. Then there exist x 2 Z N and an integer 0 <dapple N /k such that {x, x + d k } A. The proof of Theorem 2 relies on a result of the second author in [6] and is given in Section 2. Note that for p = Theorem 2 essentially recovers the quantitative dense case, i.e., Theorem. Further, as exp (log N) 9/0 N " for all " > 0 Theorem 2 shows that for essentially the same range of probability as in Nguyen s result [7], subsets of [Z N ] p of quite smaller density than in [7] already span the desired polynomial configuration. Lastly, we note that for the density (log N) 5 log log log log N =/!(N) as in Theorem 2 the term exp (log N) 9/0 cannot be improved to any function of the form o(!(n)). In particular, the condition on p in Theorem 2 cannot be improved to p N /k as in Nguyen s result. To see this suppose that Theorem 2 holds for p = (N)N /k!(n) with (N) = o(). Consider the two round exposure [Z N ] p [[Z N ] p2 =[Z N ] p with p = (N)N /k for some function o() = (N) (N). By Cherno s bound asymptotically almost surely [Z N ] p has size at least p N/2. Further, by Markov s inequality asymptotically almost surely the number of configurations {x, x + d k },0<dappleN /k,in [Z N ] p is o(p N). Thus by deleting at most one element for each configuration {x, x + d k } we obtain a subset of [Z N ] p [Z N ] p which does not contain the configuration {x, x + d k } and which has size at least p N/4 = (N)N /k /4 Indeed, the constant /5 can be improved to /4 +o() at the cost of exp (log N) 9/0 changing to exp (log N) o(),butwestatethisversionforsimplicity.

3 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS3 2 (N)N /k =2pN/!(N). As [Z N ] p has size at most 2pN asymptotically almost surely we obtain a contradiction to the fact that Theorem 2 holds for p = (N)N /k!(n). As the second part, and the main objective of this paper, we show a transference result of Theorem for pseudo-random host. It is interesting to note that Tao and Ziegler [24] proved that the polynomial Szemerédi theorem also holds in the primes. Their proof relies heavily on pseudo-random properties of the primes, thus, can be seen in the scheme mentioned above. The notion of pseudo-randomness we shall rely on in this paper is a traditional one defined through small non-trivial Fourier coe cients. Transference problems with this notion of pseudo-randomness have been studied previously for Roth s theorem [9, 3, 4]. To our best knowledge, however, there is no result known concerning transference of the Furstenberg-Sarközy theorem to this pseudo-random setting and our next result, Theorem 3, shall give the first non-trivial bound. Given a function f : Z N! C the Fourier transform of f is defined to be the function bf : Z b N! C given by f( ) b = f(x)e( x) x2z where e(x) =e 2 ix/n.thelinear bias of f, denoted by kfk u, is then given by kfk u = The result then reads as follows. sup f( ). b b06= 2Z b N Theorem 3. Let k 2 be an integer and let > > 0 be reals with + (2 k+ 3) >. Then there is a function!(n) with lim N!!(N) = such that the following holds. Let Z N be such that N and k k u apple N, and let = (N) (log log N)!(N). Then, every subset A satisfying A contains a pair x, x + d k for an integer 0 <dapple N /k. Due to Parseval s equality the the parameter in Theorem 3 controlling the pseudo-randomness of satisfies /2. One may think of = /2 as though is as pseudo-random as possible. In this case, i.e., = /2, we have that Theorem 3 is applicable as long as > 2 k+2 5. The proof of Theorem 3 combines the techniques in [8] and an unpublished note of Green which investigates transference for Roth s theorem for pseudo-random setting. The latter is in flavor similar to Green s proof of Roth s theorem in the primes [9] and an important part of the argument relies on a discrete version of a restriction type result due to Tomas [25] (see Theorem 4). Sketch of the proof of Theorem 3 and a restriction type theorem. For notational convenience we identify a set with its characteristic function, i.e., if A Z N then A also denotes a 0, -function with A(x) = if and only if x 2 A. Moreover, let Q k = {x k :0<xapple N /k,x is integer } denote the set of kth powers. Given a sparse pseudo-random set Z N and a subset A with relative density

4 4 ELAD AIGNER-HOREV AND HIỆP HÀN (with respect to ) as in Theorem 3 we will bound P A(x)A(x + d)q k (d) from below as follows. Using the large Fourier coe cients of A and Bohr sets we will construct a function a : Z N! R + which satisfies kak N and which is bounded from above, kak apple 3 (Lemma ). Hence, the function a resembles a set in the dense setting and by a Varnavides argument and Theorem we then establish a lower bound on the of number of polynomial configurations contained in a, i.e. a lower bound for P a(x)a(x + d)q k (d), (Lemma 7). Under the assumption that A does not satisfy the conclusion of the Theorems 3, we will exploit the close relationship of A and the function a to obtain an upper bound for the number of configurations contained in a which contradicts the lower bound mentioned above. The contribution of the large Fourier coe - cients of A can be controlled by the properties of Bohr sets and the contribution of the remaining small Fourier coe cients will be controlled using the following restriction type theorem which is at heart of the approach. Theorem 4. Let > > 0 and p = 2 2 be fixed real numbers. Let Z N with size N and linear bias k k u apple N. Then for every A we have 2 b Z N b A( ) p apple 2 p A p/2 p/2. () Theorem 4 is from an unpublished note of Green (see also [9] and [7]) and will be given in Section 7. It is a discrete version and a close adaptation of the restriction theorem due to Tomas [25]. It is even more closely related to the work of Mockenhaupt and Tao [5] and we refer to this work for further applications of restriction results. 2. Transference for random host Proof of Theorem 2 In this section we prove Theorem 2. For an integer k 2 recall that Q k = {x k :0< x apple N /k,x is integer}. Consider the graph G = (Z N,E k )withthe vertex set Z N in which we connect each x 2 Z N to x + d for any d 2 Q k. Then a set A Z N, which does not span the polynomial configuration {x, x + d} with d 2 Q k, corresponds to an independent set in G. Theorem then gives an upper bound on the size of the largest independent set in G and the proof of Theorem 2 will rely on the following related notion of largest almost independent set. Definition 5. Given constants 2 [0, ], 2 [0, ], and a graph G on N vertices. We say that G is a (, )-supersaturated graph if for any subset S V (G) with 2 S e(s) apple e(g), N we have S apple N. In addition, let = (n) > 0 and = (n) > 0. For a sequence of graphs G = {G n } n2n, we say that G is (, )-supersaturated if there exists a constant n 0 2 N such that for every n n 0, G n is ( (n), (n))-supersaturated. The following result is a direct consequence of Proposition 2.6 from [6]. Proposition 6 (Proposition 2.6 from [6]). Let = (n) and = (n) be (0, )- valued functions, and let G = {G n } n2n be a sequence of (, )-supersaturated

5 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS5 graphs, in which each G n has N = N(n) vertices (with lim n! N(n) =) and average degree D = D(n). Further, let V p denote the random vertex set obtained by choosing each vertex from V = V (G n ) with probability p independently at random and let H n = G n [V p ] denote the subgraph of G n induced by the vertex set V p. Then for p = p(n) ( D) log 2 (e/ ) asymptotically almost surely (H n ) apple 2 pn holds. Proposition 2.6 from [6] is actually more precise but this is insignificant in our setting. We want to apply this proposition and by using Theorem combined with a Varnavides type argument [26] we derive that the the graph (Z N,E k )isindeed supersaturated for suitable choice of parameters. We also establish the functional version which is needed for the proof of Theorem 3. Lemma 7. There exists a c>0 such that for all k 2, thefollowingholdsfor su ciently large N. Let!(n) = log log log log n and let M : R! R be given by M(x) =exp x 4./ log log log(x). Suppose that = (N) satisfies c > 2 (log N)!(N)/5. Then M(/ ) < exp 3 (log N)9/0 and for any set A Z N of size A N we have A(x)A(x + d)q k (d) 2M(/ ) 2 N Q k. (2) In particular, if c> 3(log N)!(N)/5 and a : Z N! R is a function with kak N and kak apple 3 then a(x)a(x + d)q k (d) 3 48M(6/ ) 2 N Q k. (3) Moreover, let N = N(n) be a sequence of increasing primes, then the graph sequence G =(G n ) n2n with G n =(Z N,E k ) is,m(/ ) 2 -supersaturated. Proof. We first note that (3) and the last part of the lemma follow from (2). Indeed, from Definition 5 and (2) it is immediately seen that for < /2 we have that G = (G n ) n2n is,m(/ ) 2 -supersaturated. Moreover, for a function a : Z N! R with kak N and kak apple 3 consider the set A = {x: a(x) /2}. Then we have A N/6 since otherwise kak < kak N/6+N /2 apple N. Withthejust defined set A we then have a(x)a(x + d)q k (d) 2 4 A(x)A(x + d)q k (d) and (3) follows from (2). To establish (2) we first apply Theorem to obtain a constant C. Let c>0be su ciently small and let N be su ciently large for the calculations to hold. For a given define M = M = M(/ ) which is increasing in / if is su ciently small. Moreover,!(M) = log log log log M = log log 4. log log log(/ ) log > 0.99 log log log.

6 6 ELAD AIGNER-HOREV AND HIỆP HÀN For su ciently this implies! 4./ log log log(/ )!(M)/4.0 (log M)!(M)/4 = Consequently from the choice of the function! and from Theorem any subset of [M] = {, 2,...,M} of density at least /3 C(log M)!(M)/4 contains a configuration {x, x + d}, d 2 Q k. We will use this to establish (2). Let A Z N of size A N be given. Consider the N Q k progressions in Z N given by P (x, d) ={x, x + d,..., x +(M )d} with x 2 Z N and d 2 Q k. Each progression has M elements since N is prime and we call such a progression good (with respect to A) if A \ P (x, d) M/2. For a fixed d note that P x2z N A \ P (x, d) = A M and for fixed x and d we have A\P (x, d) apple M. Hence, for a fixed d there are at least A /2 elements x 2 Z N such that P (x, d) is a good progression. Consequently, with A N we conclude that there are at least N Q k /2 good progressions in total. We identify a good progression P (x, d) with the interval {,...,M} and recall that M/2 >MC(log M)!(M)/4. By Theorem we conclude that that each good progression contains a pair {x, x + d} A, d 2 Q k. Moreover, each such pair {x, x + d} is contained in at most M 2 progressions as a progression is determined after choosing the positions of {x, x + d} in the progression. We obtain A(x)A(x + d)q k (d) N Q k /2M 2, as claimed. It is left to show that M = M(/ ) < exp 3 (log N)9/0. Let 0 =!(N)/5 2 (log N) and let M 0 = M(/ 0 ). As M(/ ) is increasing in / for small enough, it su ces show that M 0 apple exp 3 (log N)9/0, equivalently, log M 0 apple 3 (log N)9/0. For su ciently large N we have 4./ log log log(/ 0) log M 0 = = 2(log N)!(N)/5 4./ log log log(/ 0). 0 3C. As log log log / 0 > log log log log N =!(N) we obtain log M 0 < 2(log N) 4./5 < 3 (log N)9/0 for su ciently large N. This finishes the proof. With Lemma 7 at hand we now prove Theorem 2. Proof of Theorem 2. Let N = N(n) be an increasing sequence of primes. Let G = {G n } n2n be a sequence of graphs with each G n =(Z N,E k ) be defined as above. In particular, G n has N = N(n) vertices and degrees D = D(n) = Q k. We apply Lemma 7 we obtain the constant c>0. Let > (log N) 5 log log log log N be given. Due to monotonicity we may assume that c> and let = /2 > 2 (log N) log log log log(n)/5. Lemma 7 then guarantees that the graph sequence G is (, )-supersaturated with = M 2 =exp 2(/ ) / log log log /. By Proposition 6 we then derive that for any p ( D) log 2 (e/ ) asymptotically almost surely the largest independent set in H n = G n [V p ] has size at most 2 pn = pn. Further, we have log 2 e/ apple M and by Lemma 7 we also have M < exp{ 3 (log N)9/0 }. Hence ( D) log 2 (e/ ) N /k exp{(log N) 9/0 }.Thetheorem follows.

7 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS7 3. Transference for pseudo-random host Preliminaries In this section we introduce the definitions and basic properties of discrete Fourier analysis and Bohr sets as well as the theorem of Hardy-Littlewood. 3.. Fourier analysis. As shown in the introduction, for the purposes of Fourier analysis we endow Z N with the counting measure and, consequently, endow its dual group Z b N with the uniform measure. As a result, the Fourier transform of a function f : Z N! C is the function bf : b Z N! C given by b f( ) = x2z f(x)e( x) where e(x) =e 2 ix/n. Further, given g : Z N! C, letf g be the convolution of f and g defined by f g(x) = P y2z N f(y)g(x y). The basic properties of the Fourier transform then read as follows f(x) = P b N 2Z b N f( )e( x) for all x 2 ZN (Inversion), P x2z N f(x) 2 = P b N 2Z b N f( ) 2 (Parseval), [f g( ) = f( )bg( ) b (Convolution) Bohr sets. Given a set S b Z N and a real 0 apple % apple, let B(S, %) ={y 2 b Z N : (y) apple %, for all 2 S} denote the Bohr set with frequency set S, rank S, and radius %. Then, see, e.g. [23, Lemma 4.20]. A Bohr set B(S, %) is called regular provided B(S, %) % S N (4) ( 00 S apple ) B(S, %) apple B(S, (+ apple)% apple ( + 00 S apple ) B(S, %) (5) for all apple apple /00 S. Regular Bohr sets are easily found as suggested by the following. Theorem 8. (see [2] or, e.g., [23, Lemma 4.25]) Let S Z b N be nonempty and let 0 < " <. Then there exists a % 2 ["/2, "] such that B(S, %) is regular. The following standard property of Bohr sets will be useful to us. Proposition 9. If B = B(S, %) and 2 S then b B( ) apple %.

8 8 ELAD AIGNER-HOREV AND HIỆP HÀN Proof. b B( ) = = = apple (x) x2b (x) x2b ( (x)) x2b (x). As 2 S we have (x) apple % and the proposition follows Waring s problem. Given positive integers s, k and n, letr s,k (n) to denote the number of solutions (in the integers) to the equation x2b x k + x k s = n. Then r s,k (n) can be expressed as an s-fold convolution of Q k = {x k : x apple N /k,x2 Z N } as follows r s,k (n) =Q k Q {z k (n). } s times The following is a well-known result of Hardy and Littlewood [] (see also [6], Theorem 5.7) solving a well-known problem of Waring. Theorem 0. (Hardy-Littlewood []) For every s 2 k we have r s,k (n) = (n s/k ). 4. Transference for pseudo-random host A dense set model As mentioned in the introduction we will construct a model for a given a subset A. This is established by the following lemma. Lemma. Let Z N and for = (N) > 0, " = "(N) > 0 let A with A and let ;6= S Z b N and % = %(N) 2 ["/2, "] be such that B = B(S, %) is a regular Bohr set. Then the function a : Z N! R given by satisfies: a(x) = N (A B)(x) () kak N, and (2) if k k u > 2(20 S /2 /") S, then kak apple 3. Proof. The property kak N is clear and we focus on proving kak apple 3. For given S Z c N choose apple =/(00 S ) and let B = B(S, (+apple)%) and B 2 = B(S, apple%), so that B 2 B B. Since (B B 2 )(x) B 2 for all x 2 B and B B the function f(x) =(B B 2 )(x)/ B 2 satisfies f(x) B(x) for all x 2 Z N. Hence,

9 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS9 A B(x) apple f(x), for every x 2 Z N, and by Fourier-inversion and the convolution property we have a(x) apple N ( f)(x) = b ( ) f( ) b 2Z c N = b (0) b f(0) + 2 c Z N \{0} b ( ) b f( ). By the convolution property b f( ) = c B ( ) c B 2( ) B 2, hence, using Cauchy-Schwarz and Parseval, we derive a(x) apple B + k k u B B 2 c ( ) B c 2 ( ) 2Z c N apple B + k k u B 2 kc B k 2 kb c 2 k 2 = B + N k k u B /2 B 2 /2 (6) Using (5) and B / apple 2, which follows from the choice of apple, we obtain p k k u 2 a(x) apple 2+N ( B 2 ), /2 By (4) we conclude ( B 2 ) /2 N(%apple /2 ) S N("/(20 S /2 )) S > 2Nk k u k Hence, kak apple 3, as required.. We just showed that a sparse set A can be associated to a function a which behaves like a characteristic function of a dense set. By (3) of Lemma 7 this function a contains many of the desired polynomial configurations, i.e. P a(x)a(x + d)q k (d) is large. In the next section we shall work towards an upper bound for P a(x)a(x + d)q k (d). Under the assumption that the sparse set A does not contain the desired configuration this upper bound will then yield a contradiction with (3). 5. Transference for pseudo-random host An upper bound In this section we use Theorem 4 and Theorem 0 to derive an upper bound for the contribution of the small Fourier coe cients. The method of using Theorem 0 to deal with the Fourier coe cients of Q k was introduced in [8]. Lemma 2. For all k 2 there is a W such that for all > 0 and > 0 satisfying + (2 k+ 3) ( ) ( ) the following holds. Let Z N with be a set of size N with linear bias k k u apple N. Let = (N), A and S b Z N such that sup 62S b A( ) apple.

10 0 ELAD AIGNER-HOREV AND HIỆP HÀN Then we have A( ) b 2 Qk b ( ) apple W (2t p)/t N /k 2. (7) 62S Proof. Choose W such that r t/2,k (n) apple (W/4) t/2 n t/2k ; this is possible due to Theorem 0. Let t =+/(2 k+ 3) and let t 0 = t/(t ) = 2(2 k ) be the dual index of t. By Hölder inequality applied with t and t 0 we have: 0 /t A( ) b 2 Qk b ( ) apple A( ) b 2t A kq b k k t 0. 62S By Theorem 4 with p =(2 2 )/( ) and by the choice of t we have 2t >p and hence A( ) b 2t apple sup A( ) b 2t p A( ) b p apple ( ) 2t p (2 ) p. 62S 62S 62S Consequently, it su ces to show that k b Q k k t 0 apple WN /k /4. (8) To this end, recall that the (t 0 /2)-fold convolution of Q k coincide with r t/2,k. Hence, by appealing to the convolution property, Parseval and Theorem 0 we obtain Q b k ( ) t 0 = 2 bqk ( ) t0 /2 2Z c N 2 c Z N = bqk b Q k ( ) 2 2 c Z N apple N (Q k Q k (n)) 2 n2z N apple N 2 (W/4) t0 N t0 /k 2 apple (W/4) t0 N t0 /k. 6. Transference for pseudo-random host Proof of Theorem 3 We are now in the position to prove Theorem 3. Proof of Theorem 3. For given k 2 let W be the constant obtained by applying Lemma 2 with k. Let > > 0 be such that + 2 k+ 3 > and let t =+ (2 k+ 3) and p = 2 2. Then p<2t, thus,p<4 and t/(2t p) is a constant depending on, and k but independent of N. For!(N) = 0 log log log log log N let = (N) > (log log N)!(N). We may further assume that is su ciently small for the calculations to hold (say <c for some constant c>0). Let M(/ ) =exp{(/ ) 4./ log log log(/ ) }

11 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS and recall from (3) of Lemma 7 that any function a : Z N! R with kak and kak apple 3 satisfies 3 a(x)a(x + d)q k (d) 48M(6/ ) 2 N Q k. Let = 3 48 M(6/ ) 2 and = 0W t/(2t p) and " = p 0 2 p. For a su ciently small > (log log N)!(N) and su ciently large N a straightforward computation shows that (2 k ku p (2/ ) p 20) = N > 2. (9) This puts us in the position to apply Lemma as argued in the following. Given a set A with A let denote the " p S = Spec (A) ={ 2 b Z N : b A( ) } -spectrum of A. Due to Theorem 4 we have N Spec (A) apple (2/ ) p. (0) By Theorem 8 there is a % 2 ["/2, "] such that the Bohr set B = B(S, %) ={y 2 bz N : (y) apple % for all 2 S} is regular. Hence, we can apply Lemma with S and " to conclude that the function a = N (A B) satisfies kak N and kak apple 3. From (3) of Lemma 7 we obtain a(x)a(x + d)q k (d) N Q k. () Assume that A does not satisfy the conclusion of Theorem 3, (actually we will even work under the following weaker assumption that A contains only few desired configuration) A(x)A(x + d)q k (d) < 2 Q k 2 N. (2) We will derive an upper bound contradicting () as follows. Due to inversion and orthogonality of characters we have A(x)A(x + d)q k (d) = A( b N 3 ) A( b 2 ) Q b k ( 3 )e(x( + 2 ))e(d( + 3 )), 2, 32Z b N = A( ) b A( b ) Qk b ( ). (3) N 2Z b N Hence, (2) translates to 2 N Q k N 2 2 b Z N b A( ) b A( ) b Qk ( ) > 0. (4)

12 2 ELAD AIGNER-HOREV AND HIỆP HÀN Further, by the definition of a we obtain due to the convolution property that a(x)a(x + d)q k (d) = ba( )ba( ) Q N b k ( ) 2Z b N = N 2 2 b Z N b A( ) b A( ) b Qk ( ) bb( ) b B( ) 2. (5) Hence, adding the left hand side of (4) to the right hand side of (5) we obtain a(x)a(x + d)q k (d) < N 2 2 b Z N b A( ) 2 b Q k ( ) To derive a contradiction to () it is now su 2 b Z N b A( ) 2 b Q k ( ) b B( ) 2 2 cient to show that b B( ) 2 2 apple 2 Q k / N Q k. To this end, we split the sum on the left hand into one over 2 S = Spec (A) and another over 62 S. To estimate the first sum, recall from (0) that S apple (2/ ) p and from Proposition 9 we know for all 2 S b B( ) We now conclude that A( ) b 2 Q b k ( ) 2S apple % hence b B( ) 2 2 apple 2%. b B( ) 2 2 apple 2% S b A(0) 2 b Q k (0) apple 2 Q k /4 due to the choice of " and % 2 ["/2, "]. Using (7) of Lemma 2 we derive that the sum over 62 S is at most 2 A( ) b 2 Q b k ( ) apple 2W (2t p)/t 2 Q k apple 2 Q k /4 2 b Z N \S due to the choice of. This concludes the proof. Remark 3. Note that the proof indeed implies that a set A in the Theorem 3 contains at least 2 2 Q k N = 3 96 M(6/ ) 2 N of density as 2 N Qk configurations of the form x, x + d, d 2 Q k. Up to the the factor this bound is best possible. 7. Arestrictiontheorem-ProofofTheorem4 In this section we prove Theorem 4. We first introduce some notation. Let Z N be endowed with the counting measure µ and its dual Z b N be endowed with the normalised counting measure. Given a function g : Z b! C we define the inverse Fourier of g to be g _ determined by Z g _ (x) = g( )e( x)d = g( )e( x). (6) 2Z b N N 2Z b N

13 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS 3 Given a subset Z b N we endow with the induced measure, i.e. the normalised counting measure ( ) = ( )/ on. Thus, has total mass and we define the restricted Fourier transform on via (gd ) _ (x) = 2 b Z N g( ) ( )e( x). (7) Let B(Z N,µ) denote the space of all functions f : Z N! C. We are interested in the linear operator called restriction map T : B(Z,µ)! B(, ) given by f 7! b f. It is easy to check that its adjoint operator is T : B(, )! B(Z,µ) given by g 7! (gd ) _. Lastly, let kt k r!q be the infimum of all C such that ktfk Lq (Z N,dµ) apple Ckfk Lr (,d ) for all f 2 B(, ). The main objective of this section is to establish the following. Theorem 4. Let p and T be defined as above. Then kt k 2!p apple 2, i.e. for all f 2 B(, ) we have ktfk Lp (Z N,µ) apple 2kfk L2 (, ). With this result Theorem 4 easily follows. Proof of Theorem 4. Recall that Z N is isomorphic to its dual Z b N and that g _ (x) = bg( x). With this in mind let A Z b N be given. Then (Ad ) _ (r) = 2 b Z N A( )e(r ) = ba( r), from which we derive k(ad ) _ k p L p (Z N,µ) = p/2 p A( b r) p apple 2 p kak p A L 2 (, ) =2p r2z and the theorem follows. The proof of Theorem 4 will utilise the following principle kt k 2 2!p = kt k 2 p 0!2 = ktt k p0!p

14 4 ELAD AIGNER-HOREV AND HIỆP HÀN where p 0 is the dual index of p, i.e.p 0 =(2 2 )/(2 ). Indeed, we will need kt k 2 2!p applektt k p 0!p only. This is easily seen by using Hölder s inequality ktfk p = sup kgk p 0 = htf,gi = sup hf,t gi kgk p 0 = applekfk 2 =kfk 2 applekfk 2 sup kgk p 0 = sup kgk p 0 = sup kgk p 0 = kt gk 2 =kfk 2 ktt k /2 p 0!p hg, TT gi /2 kgk /2 p 0 ktt gk /2 p Further, we will utilise the following interpolation theorem, see [7]. Theorem 5 (Riesz-Thorin interpolation theorem). Let L : B()! B(Y ) be a linear operator and suppose that p 0,p,q 0,q 2 [, ] satisfy p 0 <p and q 0 <q. For any t 2 [0, ] define p t and q t by Then = t + t and p t p 0 p = t + t. q t q 0 q klk pt!q t appleklk t p 0!q 0 klk t p!q. We are now in the position to prove Theorem 4. Proof of Theorem 4. Note that the map As (d ) _ (r) expands to TT : B(Z N,µ)! B(Z N,µ) is given by f 7! f (d ) _ (d ) _ (r) = ( )e(r ) = b ( r) we have (d ) _ (0) = and, by the pseudorandomness assumption, (d ) _ (r) apple N for r 6= 0. We define K =(d ) _ 0,whichis(d ) _ with the origin removed. Then K satisfies kkk apple N and kkk b = max Nd ( ) apple N. To show ktt k p 0!p apple 2, i.e. kf 0 + f Kk L p (Z N,µ) apple 2kfk L p 0 (Z N,µ) we first note that kf 0 k Lp (Z N,µ) applekfk L p 0 (Z N,µ) as p p0. To finish the proof we need to establish kf Kk L p (Z N,µ) applekfk L p 0 (Z N,µ). (8) To do so, we will use the Riesz-Thorin interpolation theorem, Theorem 5, to interpolate between L -L and L 2 -L 2 norm. For the former we obtain the bound kf Kk applekkk kfk L (Z N,µ) apple N kfk L (Z N,µ).

15 POLYNOMIAL CONFIGURATIONS IN SUBSETS OF RANDOM AND PSEUDO-RANDOM SETS 5 A bound for the latter follows from Plancherel and k b Kk apple N : kf Kk L 2 (Z N,µ) = k b f b Kk L2 ( b Z N, ) applekb Kk k b fk L2 ( b Z N, ) apple N kfk L 2 (Z N,µ). We choose q 0 =, p 0 =, q = 2, p = 2 and t = 2 [0, ] and apply the Riesz-Thorin interpolation theorem with the linear operator f 7! f K to obtain and ( t) = + t = p t p 0 p p 0 and ( t) = + t = q t q 0 q p kf Kk Lp (Z N,µ) apple N ( )( t) N ( )t kfk L p 0 (Z N,µ) = kfk L p0 (Z N,µ) which establish (8) and the theorem follows. References. A. Balog, J. Pelikán, J. Pintz, and E. Szemerédi, Di erence sets without appleth powers, Acta Math. Hungar. 65 (994), no. 2, J. Bourgain, On triples in arithmetic progression, Geom.Funct.Anal.9 (999), no. 5, D. Conlon and W.T. Gowers, Combinatorial theorems in sparse random sets, submitted. 4. David Conlon, Jacob Fox, and Yufei Zhao, Extremal results in sparse pseudorandom graphs, submitted. 5. Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J.AnalyseMath.3 (977), M. Gauy, H. Hàn, and I. Oliveira, Erdos-Ko-Rado for random hypergraphs: asymptotics and stability, submitted. 7. Ben Green, Restriction and Kakeya phenomena, notesfromacoursegiveninpartiiiofthe Cambridge Mathematical Tripos (2002). 8. Ben Green, On arithmetic structures in dense sets of integers, DukeMath.J.4 (2002), no. 2, Ben Green, Roth s theorem in the primes, Ann.ofMath.(2)6 (2005), no. 3, M. Hamel, N. Lyall, and A. Rice, mproved bounds on sarkozy s theorem for quadratic polynomials, Int.Math.Res.Notices(203),no.8, G. H. Hardy and J. E. Littlewood, A new solution of Waring s problem, Q. J. Math. 48 (99), Balogh József, Morris Robert, and Wojciech Samotij, Independent sets in hypergraphs, submitted. 3. Yoshiharu Kohayakawa, Vojtěch Rödl, Mathias Schacht, and Jozef Skokan, On the triangle removal lemma for subgraphs of sparse pseudorandom graphs, Anirregularmind,BolyaiSoc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 200, pp I. Laba and M. Hamel, Arithmetic structures in random sets, Integers: Elec.J.ofcombin. num. th. 8 (2008), Gerd Mockenhaupt and Terence Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J. 2 (2004), no., Melvyn B. Nathanson, Additive number theory, GraduateTextsinMathematics,vol.64, Springer-Verlag, New York, 996, The classical bases. 7. H. H. Nguyen, On two-point configurations in random set, Integers9 (2009), János Pintz, W. L. Steiger, and Endre Szemerédi, On sets of natural numbers whose di erence set contains no squares, J.LondonMath.Soc.(2)37 (988), no. 2, I. Z. Ruzsa, Di erence sets without squares, Period.Math.Hungar.5 (984), no. 3, A. Sárkőzy, On di erence sets of sequences of integers. I, ActaMath.Acad.Sci.Hungar.3 (978), no. 2, David Saxton and Andrew Thomason, Hypergraphs containers, submitted. 22. M. Schacht, Extremal results for random discrete structures, submitted. 23. T. Tao and V. H. Vu, Additive combinatorics, CambridgeStudiesinAdvancedMathematics, vol. 05, Cambridge University Press, Cambridge, 200.

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