A Remark on Sieving in Biased Coin Convolutions

Transcription

1 A Remark on Sieving in Biased Coin Convolutions Mei-Chu Chang Department of Mathematics University of California, Riverside Abstract In this work, we establish a nontrivial level of distribution for densities on {,..., N} obtained by a biased coin convolution. As a conseuence of sieving theory, one then derives the expected lower bound for the weight of such densities on sets of almost primes. Introduction. Over the recent years, there has been an increasing interest in sieving problems in combinatorial objects without a simple arithmetic structure. The typical example is that of finitely generated thin subgroups of linear groups such as SL Z) or SL Z + iz). These groups are combinatorially defined but are not arithmetic they are of infinite index) and as such cannot be studied with classical automorphic techniues. Examples of natural appearances of this type of uestions include the study of the curvatures in integral Apollonian circle packings, Pythagorean triples and issues around fundamental discriminates of uadratic number fields and low lying geodesics in the modular surface. See [].) The reader may also wish to consult the excellent Bourbaki exposition by E. Kowalski [6] for a detailed account of many of these recent developments around exotic sieving. 00 Mathematics Subject Classification.Primary 60B99. Key words. random polynomial, double root, coin convolution, sieving. Research partially financed by the NSF Grants DMS

2 In this paper we consider a slightly different problem but in a somewhat similar spirit. Let N = m and identify {,..., N} with the Boolean cube {0, } m through binary expansion. Denote µ ρ the probability measure on {0, } m given by a standard biased coin convolution, i.e. on each factor we take an independent distribution assigning probability ρ to 0 and ρ to. Consider the resulting distribution on {,..., N}. For ρ =, this is the uniform distribution while for ρ, these distributions become increasingly singular. Our aim is to study some of their arithmetical properties and in particular prove that there is a nontrivial level of distribution no matter how close ρ is to, ρ <. Similar results may also be obtained for g-adic analogues, expanding integers in base g. Notations. Throughout the paper, any implied constants in the symbols O,, and may depend on the parameter ρ > 0. We recall that the notations A = OB), A B and B A are all euivalent to the statement that the ineuality A cb holds with some constant c > 0, and that A B means A is eual to B asymptotically. The distance from x to the nearest integer is denoted by x. For brevity, we set ) θ eθ) = e πiθ, e θ) = e. The constant c may vary in the same statement and depend on the parameter ρ. The statement. Consider the distribution µ on [, N] Z, with N = m, induced by the random variable j ξ j j with ξ j ), j 0, be an independent, identically distributed seuence of random variables taking values in {0, }, Pξ j = 0) = ρ, Pξ j = ) = ρ, < ρ <. Thus, if n = j a j j with a j {0, } the binary expansion, then µn) = ρ m l ρ) l, where l = j a j.) Note that for ρ = we obtain the normalized uniform measure on [0, N].

3 The measure.) has dimension ρ) log and hence becomes more ρ irregular for ρ. Our aim is to establish a level of distribution of µ in the sense of sieving theory. Thus, taking < N α, suare free and α appropriately small, since µ is normalized) we may write where µ [ n N : n ] = R = λ=0 n= = + R, λ= n= N e λn)µn) N e λn)µn). We also assume odd. The number α is the sieving exponent [5]. Our aim is to obtain a bound of the form ) R = o log N <N α.).3) where sums over suare free and odd. See 4.) for a better bound.) Theorem. Let the notations be as above. Then µ has sieving exponent αρ) > 0. In fact, αρ) = O ρ) for ρ. Theorem will be proved in Section 4. Theorem provides the necessary ingredient to apply sieving theory and in order to obtain almost primes integer whose prime factors are bounded below) satisfying certain property. More precisely, from the standard combinatorial sieve, we also apply to measure instead of set. See e.g. Corollary 6. in [4]. Also, [], [], [3].) We have the following result. Corollary. Let µ be the probability distribution defined as above, and let P N,α = {n N : the prime factors of n are at least N α 0 }. Then there are constants 0.8 > C > C > 0.36 such that for sufficiently large N we have See 4.7) for C and C.) C α log N < µp N,α) < C α log N..4) 3

4 Remark. If n N has no prime factors less than N α 0, then n has at most [0/α] prime factors. Since α is small, the number of prime factors in the almost primes is rather large. First estimates. Let Therefore, Then <N α R R = = R <N α λ= n= λ= j<m N e λn)µn) λ= j<m <N α log N α <N α ρ + ρ) e λ j )). ) λ ρ + ρ) e..) λ = j<m,λ )= λ= j<m,λ)= ) ρ + ρ) e λ ) λ ρ + ρ) e..) Define Note that R = λ= j<m,λ)= ) λ ρ + ρ) e..3) ρ + ρ)eθ) = 4ρ ρ) sin πθ..4) Let us consider first the case of small. 4

5 For λ 0 mod, identity.4) implies ) λ ρ + ρ) e c for c > 0 so that R < ) c m < e c m < N c/. One can do better by the following observation. Let I {,..., m} be an arbitrary interval of size I = [log ] + and let λ Z be an integer satisfying λ 0 mod. Claim. ) max λ j I sin π > c,.5) where c > 0 is some constant independent of. Proof of Claim. Let I = {j 0, j 0 +,..., j 0 + s} with s = [log ] and let ξ = λ j 0 The assumptions that λ 0 mod and that is odd imply ξ 0, and hence ξ. Fix a constant c sufficiently small. Assume ξ < c. It is clear that for s N small, λ j 0 +s = λ j 0 +s = s ξ. It is also clear that λ j 0 +s > c for some s s. Otherwise, s c c, which contradicts to the assumption ξ s log. Now, the claim follows from the fact that sinθπ) θ π for θ small. Therefore, dividing the index interval [, m] in.3) in subintervals of length log, by.4) and.5), we also have R < cρ) ) m [log ]+ < N cρ) [log ]+ < e cρ) log N.6) if log = O log N ). To obtain the first ineuality, in each subinterval of j we use.5) for the factor with sin λ j π) achieving maximum, and the trivial bound for the other factors.) Remark. In.6) we can make cρ) explicit by choosing s and c explicitly, and using the ineuality sinθπ) > θ π for θ small in the proof of the Claim. More precisely, if we take s = [log ] + and c =, we can take 0 cρ) = ρ ρ)π/0 in.6). 5

6 3 Further estimates. We want to estimate Q <Q with Q < N α and log Q log N. We will show that R Q 3.) Q <Q To show 3.) we may assume α = t for some large even integer t > given in 3.7) ). Choose h Z such that R h Q < h+. 3.) Since Q < N α, we have h < t m < m. Estimate 3.) using Hölder ineuality R Q <Q Q <Q Q <Q Q <Q t/ τ= t/ τ= [ [ Q λ=,λ)= Q t [ t/ Q τ= Q <Q Q <Q Q Q t/ τ= [ λ=,λ)= λ=,λ)= λ=,λ)= λ=,λ)= τh j=τ )h t/ τ= τh j=τ )h τh j=τ )h ) λ ρ + ρ) e τh j=τ )h ) λ t/] /t ρ + ρ) e ) λ t/] /t ρ + ρ) e ) λ t/] /t ρ + ρ) e h ) λ τ )h ρ + ρ) e t/] /t 6

7 Q <Q Q λ =,λ )= h λ ρ + ρ) e j ) t/. 3.3) To finish the estimate, we need the following two lemmas. Lemma 3. For all θ, 0 < δ < e l > and log δ ρ ρ), 3.4) we have ρ + ρ)eθ) l δ) sin πθ. 3.5) Proof. Let γ = 4ρ ρ) sin πθ. By.4), ρ + ρ)eθ) l = γ) l. We consider the following two cases. i). γ > log. l δ Then γ) l e lγ < δ < δ) sin πθ. ii). γ l log δ. Let l = l log δ < l, and estimate γ) l < γ) l < e l γ < l γ lρ ρ) = log δ < sin πθ sin πθ < δ) sin πθ. Note that the third ineuality is because l γ = lγ log δ.) 7

8 Lemma 4. Let γ be positive. Then for all θ and 0 < δ <, we have δ) sin θ < + γ δ) sin θ + γ). 3.6) Proof. Using the identity sin A sin B = sina + B) sina B) on the difference of both sides of 3.6), we obtain γ δ) sinθ + γ) sin γ γ δ)γ > 0. Let t > 4 log δ ρ ρ). 3.7) With θ = λ j /, Lemma 3 implies that 3.3) is bounded by Q Q <Q λ=,λ)= h δ) sin πλ j ) ). 3.8) First, we will replace the discrete sum Q <Q For Q given, let { λ S = λ< λ,)= } : λ <, Q < Q, λ, ) = [0, ]. by integral. We note that S is Q )Q ) separated. We recall that a set S is H- separated if for all distinct x, y S, x y H.) In Lemma 4, taking γ = π j β with β [0, β] for some β = O h ) to be specified later, we bound 3.8) by Q λ S h + γ δ) sin π j λ + β ) )). 3.9) 8

9 We will use integration to bound 3.9) by replacing S by S β = S + [0, β]. Since S β is disjoint union of subintervals in [0, ], averaging over β [0, β] gives h + γ δ) sin π j x) ) dx βq S β βq 0 More precisely, we take h + γ δ) sin π j x) ) dx. 3.0) which implies γ < δ) and bound 3.0) by 4 δ Q 0 β = δ 4 Q, 3.) h + δ δ) sin π j x) ) dx = 4 δ Q + δ δ = 4 ) h + 3δ δ Q ) h Q /, 3.) for δ small enough such that + 3δ < 4 ). Putting 3.3), 3.8)-3.0) and 3.) together, we obtain the intended bound on 3.). 4 The proofs. Proof of Theorem. We will use the estimates.6) and 3.) to prove.3). First, we will 9

10 bound R <N α. R = R + R <N α log <c log N log c log N <e c log N e c log N + j ) log N =e c log N + e c log N. j e c e c log N ) ) N α ) 4.) Here we take c sufficiently small, e.g. c = ρ ρ)c c log From.) and.3), R R log N α R <N α <N α <N α which goes to 0 as N goes to infinity. From 3.7), α = t Hence α = O ρ) for ρ. log N α e c log N < < ρ ρ) 4 log δ. with c given in.5).) ) o, log N Corollary follows immediately from Theorem and Corollary 6. in [4]. For the readers convenience we will include the statement of the latter here. First, we introduce some notations. Let {a n : n N} R 0, P z) = p, Sx, z) = p<z, p prime n x, n,p z))= a n and Assume A d x) = a n. n x, n 0 mod d A d x) = a n + r d x). 4.) d n x 0

11 Then we denote V z) = Rx, y) = Let K > and κ > 0 satisfy w p<z p P z) d<y, d P z) ) p p) K r d x), y < N α and let β = 9κ +, any s β. The following is Corollary 6. in [4]. Proposition. [IK] Let notations be as above. Then p<z ) κ log z 4.3) log w Sx, z) < + e β s K 0 )V z)x + Rx, z s ) 4.4) Sx, z) > e β s K 0 )V z)x Rx, z s ) 4.5) Proof of Corollary. Since p ) e p<z p < ) < e p<z p p and lim z p<z p<z ) log logz) = 0.6, p p= log p ) , we have e 0.95 log z < V z) < e 0.6 log z. 4.6) So in 4.3), we may take κ =, K = e Hence β = 0, s = 0. 0 In the Proposition, we take x = N and z = N α. Therefore, RN, N α ) <N R α, and the constants in 4.4) and 4.5) are.99/e) 0 )/e 0.95 = and +.99/e) 0 )/e 0.6 = Hence we obtain the following bounds α log N o) log N < µp N,α) < α log N + o) log N. 4.7) We recall that α = t with t = tρ) given in 3.7).)

12 5 Random polynomials with coefficients in {0,, }. The initial motivation for this work came from [7], where one considers biased coin convolution densities for ternary expansions, with probabilities Pξ = 0) = ρ 0, Pξ = ) = ρ, Pξ = ) = ρ and ρ 0 ρ, ρ. The main problem focused in [7] is to ensure that the set of integers {n < N : n for some > Q} carries small weight for Q, which they manage to ensure if is not too large. The natural problem is whether such restriction is necessary. Clearly, this issue may be rephrased as the sieving problem for suare free integers, but with unrestricted level of distribution. The large values of are indeed the problematic ones.) While we are unable to provide a definite answer to their uestion and the main result of this note does not directly contribute, we will point out a simple probabilistic argument leading to the replacement of their condition. Our argument uses virtually no arithmetic structure. Let ξ j ), j 0, be an independent, identically distributed seuence of random variables taking values in {, 0, }. Let m and define the random polynomial P by P z) := m ξ j z j. In [7], the authors assumed that max Pξ 0 = x) < = ) x {,0,} 3 and proved that PP has a double root) = PP has, 0 or as a double root) up to a om ) factor, and lim m PP has a double root ) = Pξ 0 = 0). One of the open problems they raised at the end of the paper asked whether it is necessary to have assumption 5.), which enters into the proof mainly through Claim. in their paper which is crucial to their results). In this note, we will prove Claim. under a weaker assumption than assumption 5.). More precisely, we prove the following. Assume max Pξ 0 = x) < ) x {,0,}

13 Then there exist constants C, c > 0 such that for any B > 0 we have PP 3) is divisible by k for some k B) CB c. 5.3) Remark. The bound in 5.) is the solution to euation 5.). Proof. Fix r such that 3 r B < 3 r+. 5.4) Claim. PP 3) is divisible by k for some k [B, B]) cr 5.5) for some constant c > 0. Proof of Claim. We write P 3) = ξ j 3 j + j<r m ξ j 3 j. j=r Fix ξ r,..., ξ m, and let l = m j=r ξ j3 j. If k divides P 3), then ξ j 3 j l mod k. j<r Since j<r ξ j3 j < 3 r / k /, we may denote lk) := ) k ξ j 3 j, k j<r and let It follows that S = { lk) : k [B, B] } B, B ). PP 3) is divisible by k for some k [B, B]) P ξ j 3 j S ). 5.6) j<r Let σ k) = σ k) j) ),...,r {, 0, }r be defined by σ k) j)3 j = lk) j<r 3

14 and let A = {σ k) : k [B, B]} with A 3 r. Let δ j be the indicator function of j, j =, 0,, and denote ρ j := Pξ 0 = j) for j =, 0,, and ρ := max ρ j. j Denote the product measure on {, 0, } r by r ν := ρ 0 δ 0 + ρ δ + ρ δ ). Therefore we have the following. The reasoning is given below the display.) P ξ j 3 j S ) νσ) σ A j<r A /p σ A νσ) ) /, with p + = 5.7) 3 r/p ρ 0 + ρ + ρ ) r/ 3 r/p ρ + ρ) ) r/. The second ineuality is by Hölder, and the third ineuality follows from the following estimate. r ρ0 δ 0 σj)) + ρ δ σj)) + ρ δ σj)) ) σ A νσ) = σ A r = σ A a+b+c=r ρ 0δ 0 σj)) + ρ δ σj)) + ρ δ σj)) ) r a ) r a b ) ρ a 0 ρ b ρ c = ρ 0 + ρ + ρ ) r. To finish the proof of the claim, we want to show 3 r/p ρ + ρ) ) r/ < cr for some constant c > 0 5.8) i.e. 3 /p ρ + ρ) ) / <, 4

15 and we want to solve Let u = p t + t) = and rewrite 5.9) as ) p 3, with p + Let p go to infinity hence u goes to 0). Then t +u + t) +u =. 5.9) t +u + t) +u) /u = 3 5.0) = t + u log t + Ou )) + t) + u log t) + Ou )) = + t log t + t) log t) ) u + Ou ). Hence 5.0) becomes + t log t + t) log t) ) ) /u u + Ou ) =. 3 In the limit for u 0, we obtain e t log t+ t) log t) = 3. Solving t t t) t = 3, 5.) we obtain t = Putting 5.6), 5.7) and 5.8) together gives the claim. Using the same argument obtaining the second euality in 4.), we derive 5.3). It is possible to exploit somewhat better arithmetical features of the distribution under considerations but gains turn out to be minimal from 0.765), therefore, will not be elaborated here. Acknowledgement. The author would like to thank Gwoho Liu for computer assistance and I. Shparlinski and the referee for valuable comments. 5

16 References [] J. Bourgain, A. Gamburd, P. Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math., 793), , 00). [] J. Bourgain, A. Kontorovich, On the Local-Global Conjecture for integral Apollonian gaskets, Invent. Math., 04). arxiv: [3] J. Friedlander, H. Iwaniec, Opera de cribro, Amer. Math. Soc., Providence, RI, 00). [4] H. Iwaniec, E. Kowalski, Analytic number theory, Amer. Math. Soc., Providence, RI, 004). [5] A. Kontorovich, Levels of Distribution and the Affine Sieve, Annales de la Faculté des Sci. Toulouse, To appear, 04) [6] E. Kowalski, Sieve in Expansion, Séminaire Bourbaki, 63ème année, no 08, 00). [7] R. Peled, A. Sen, O. Zeitouni, Double roots of random Littlewood polynomials, preprint, 04). arxiv: