A Remark on Sieving in Biased Coin Convolutions

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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 consequence of sieving theory, one then derives the expected lower bound for the weight of such densities on sets of pseudo-primes. At the end, we make a comment on the recent paper of Peled, Sen and Zeitouni (which largely motivated this work) and show how some restrictions in their main results can be relaxed. 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 2 (Z) or SL 2 (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 techniques. Examples of natural appearances of this type of questions include the study of the curvatures in integral Apollonian circle packings, Pythagorean triples and issues around fundamental discriminates of quadratic number fields and low lying geodesics in the modular surface. (See [2].) The reader may also wish to consult the 200 Mathematics Subject Classification.Primary 60B99. Key words. random polynomial, double root, coin convolution, sieving. Research partially financed by the NSF Grants DMS

2 excellent Bourbaki exposition by E. Kowalski [6] for a detailed account of many of these recent developments around exotic sieving. In this paper we consider a slightly different problem but in a somewhat similar spirit. Let N = 2 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 2 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. In fact, the initial motivation for this work came from a paper of Peled, Sen and Zeitouni [7], where one considers biased coin convolution densities for ternary expansions, with probabilities P[ξ = 0] = ρ 0, P[ξ = ] = ρ, P[ξ = ] = ρ and ρ 0 ρ, ρ. The main problem focused in [7] is to ensure that the set of integers {n < N : q 2 n for some q > Q} carries small weight for Q, which they manage to ensure if q is not too large. The natural problem is whether such restriction is necessary. Clearly, this issue may be rephrased as the sieving problem for square free integers, but with unrestricted level of distribution. (The large values of q are indeed the problematic ones.) While we are unable to provide a definite answer to their question 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 by the weaker one max(ρ 0, ρ, ρ ) < 3 = max(ρ 0, ρ, ρ ) < t = with t the largest root of the equation t t ( t) t = 3. The argument uses virtually no arithmetic structure. It turns out to be possible to exploit somewhat better arithmetical features of the distribution 2

3 under considerations but gains turn out to be minimal ( from 0.765), therefore, will not be elaborated here. Notations. e(θ) = e 2πiθ, e q (θ) = e( θ q ). c, C = various constants. A B and A = O(B) are each equivalent to that A cb for some constant c. If the constant c depends on a parameter ρ, we use ρ. Otherwise, c is absolute. The statement. Consider the distribution on [, N] Z, with N = 2 m, induced by the measure µ(n) = j<m ξ j (n)2 j (.) with (ξ j ), j 0, be an independent, identically distributed sequence of random variables taking values in {0, }, P[ξ j = 0] = ρ, P[ξ j = ] = ρ, < ρ <. (ρ = is the normalized uniform measure on [0, N].) 2 2 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 q < N α, q square free and α appropriately small, (since µ is normalized) we may write µ [ n N : q n ] = q q λ=0 n= = q + R q, N e q (λn)µ(n) (.2) where R q = q q λ= n= N e q (λn)µ(n). We also assume q odd. The number α is the sieving exponent. Our aim is to obtain a bound of the form q<n α R q = o() (.3) 3

4 where sums over q square free and odd. Theorem. µ has sieving level α(ρ) > 0. In fact, α(ρ) = O( ρ) for ρ. From standard combinatorial sieve (which also applies to measures instead of sets.) (See e.g. [], [2], [3], [4]) we have the following. Corollary 2. µ(p r [0, N]) log N with P r = {r-pseudo-primes}, r = r(ρ). (.4) 2 First estimates. Let Note that R q = q = q q λ= n= q λ= j<m N e q (λn)µ(n) ( ρ + ( ρ) e ( λ2 j q )). (2.) ρ + ( ρ)e(θ) 2 = 4ρ( ρ) sin 2 πθ. (2.2) Let us consider first the case of small q. For λ 0 mod q, (2.2) implies q ρ + ( ρ) e ( λ2 j ) c q 2 for c > 0 so that (2.)< ( O( ) ) m C q < e m 2 q 2 < N c/q2. One can do better by the following observation. Let I {,..., m} be an arbitrary interval of size log q. λ 0 mod q, Then for { } max sin 2 λ2j q π : j I > c (2.3) with c > 0 some constant independent of q. Therefore, we also have (2.) < ( c(ρ) ) m log q < N c(ρ) log q < e log N (2.4) 4

5 if log q < O ( log N ). 3 Further estimates. We want to estimate R q (3.) q Q with Q < N α and log Q log N. It will suffice to show that (3.)< Q c for some c > 0. We may assume α = for some large t Z ( given in (3.7) ). Choose t h Z such that 2 h Q 2. (3.2) Hence h < 2 t m < m. Estimate (3.) using Hölder inequality = R q q Q q Q q Q q Q Q q λ= [ t/2 Q Q τ= q λ= t/2 τ= q λ= τh j=(τ )h τh j=(τ )h h ( λ2 j ρ + ( ρ) e q ( ) λ2 j ρ + ( ρ) e q ( ) λ2 j t/2] 2/t ρ + ( ρ) e q ) t/2. (3.3) For the last equality, we note that for each τ. {λ2 j mod p : (τ )h j < τh} ={λ2 j mod p : 0 j < h} To finish the estimate, we need the following two lemmas. 5

6 Lemma 3. For all θ, 0 < δ < and l > log δ ρ( ρ), (3.4) we have ρ + ( ρ)e(θ) 2l ( δ) sin 2 πθ. (3.5) Proof. Let γ = 4ρ( ρ) sin 2 πθ. By (2.2), ρ + ( ρ)e(θ) 2l = γ. We consider the following two cases. (i). γ > log. l δ Then ( γ) l e lγ < δ < ( δ) sin 2 πθ. (ii). γ l log δ. Let l = l 2 log δ < l and estimate ( γ) l < ( γ) l < e l γ < 2 l γ lρ( ρ) = log δ < sin 2 πθ sin 2 πθ < ( δ) sin 2 πθ. (Note that the third inequality is because l γ < 2.) Lemma 4. Let γ < /0 be positive. Then for all θ and 0 < δ <, we have ( δ) sin 2 θ + γ ( δ) sin 2 (θ + γ). (3.6) 6

7 Proof. Using the identity sin 2 A sin 2 B = sin(a + B) sin(a B) on the difference of both sides of (3.6), we obtain which is bounded by γ. Let ( δ) ( sin(2θ + γ) sin γ ), t > 4 log δ ρ( ρ). (3.7) With θ = λ2 j /q, Lemma 3 implies that (3.3) is bounded by Given Q, let Q q Q S = q λ= h ( ( δ) sin ( 2 πλ2 j ) ). (3.8) q { } λ q : 0 λ < q, q Q [0, ]. We note that S Q 2 and S is Q 2 2 h separated. In Lemma 4, taking γ = π2 j β with β [0, β] for some β = O(2 h ) to be specified later, we bound (3.8) by Q λ q S h ( + γ ( δ) sin 2 ( π2 j ( λ q + β ) )) (3.9) We will use integration to bound (3.9) by replacing S by S β = S + [0, β]. Averaging over β [0, β] gives βq S β βq 0 h ( + γ ( δ) sin 2 (π2 j x) ) dx h ( + γ ( δ) sin 2 (π2 j x) ). (3.0) dx More precisely, we take β = δ 4 Q 2, (3.) 7

8 (which implies γ < δ) and bound (3.0) by 4 δ Q 0 h ( + δ ( δ) sin 2 (π2 j x) ) dx = 4 ( δ Q + δ δ 2 = 4 ( ) h + 3δ δ Q 2 < Q /2, ) h (3.2) for δ small enough. Putting (3.3), (3.8)-(3.0) and (3.2) together, we obtain the intended bound on (3.). 4 On a paper of Peled, Sen and Zeitouni. Let (ξ j ), j 0, be an independent, identically distributed sequence of random variables taking values in {, 0, }. Let m and define the random polynomial P by m P (z) := ξ j z j In [7], Peled, Sen and Zeitouni assumed that max P(ξ 0 = x) < = (4.) x {,0,} 3 and proved that P(P has a double root ) = P(P has, 0 or as a double root ) up to a o(m 2 ) factor, and lim m P(P has a double root ) = P(ξ 0 = 0) 2. One of the open problems they raised at the end of the paper asked whether it is necessary to have assumption (4.), which enters into the proof mainly through Claim 2.2 in their paper (which is crucial to their results). In this note, we will prove Claim 2.2 under a weaker assumption than assumption (4.). More precisely, we prove the following. Assume max P(ξ 0 = x) < (4.2) x {,0,} 8

9 Then there exist constants C, c > 0 such that for any B > 0 we have P(P (3) is divisible by k 2 for some k B) CB c. (4.3) Remark. The bound in (4.2) is the solution to equation (4.0). Proof. Fix r such that 3 r B 2 < 3 r+. (4.4) Claim. P(P (3) is divisible by k 2 for some k [B, 2B]) 2 cr (4.5) 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 n, and let l = m j=r ξ j3 j. If k 2 divides P (3), then ξ j 3 j l mod k 2. j<r Since j<r ξ j3 j < 3 r /2 k 2 /2, we may denote l(k) := ( ) k ξ j 3 j 2 2, k2 2 j<r and let It follows that S = { l(k) : k [B, 2B] } ( 2B 2, 2B 2). the left-hand-side of (4.5) P ( ξ j 3 j S ). (4.6) j<r Let σ (k) = ( σ (k) (j) ) {, 0,,...,r }r be defined by σ (k) (j)3 j = l(k) j<r 9

10 and let A = {σ (k) : k [B, 2B]} with A 3 r. Let δ j be the indicator function of j, j =, 0,, and denote ρ j := P(ξ 0 = j) for j =, 0,, and ρ := max ρ j. j Denote the product measure on {, 0, } r by Therefore we have r ν := (ρ 0 δ 0 + ρ δ + ρ δ ). (4.6) σ A ν(σ) A /p ( σ A ν(σ) q ) /q, with p + q = ) r/q 3 r/p( ρ q 0 + ρ q + ρ q 3 r/p( ρ q + ( ρ) q) r/q <2 cr for some constant c > 0 (4.7) The second inequality is by Hölder, and the third inequality follows from the following estimate. r ( ρ0 δ 0 (σ(j)) + ρ δ (σ(j)) + ρ δ (σ(j)) ) q σ A ν(σ) q = σ A r = σ A a+b+c=r ( ρ q 0δ 0 (σ(j)) + ρ q δ (σ(j)) + ρ q δ (σ(j)) ) ( r a )( r a b ) ρ aq 0 ρ bq ρ cq = (ρ q 0 + ρ q + ρ q ) r To finish the proof of the claim, we want to show (4.7) < 2 cr constant c > 0, i.e. ( 3 /p ρ q + ( ρ) q) /q <, for some 0

11 and we want to solve Let u = p t q + ( t) q = and rewrite (4.8) as ( ) p 3, with p + q =. (4.8) ( t +u + ( t) +u) /u = 3 (4.9) Let p go to infinity (hence u goes to 0). Then t +u + ( t) +u = t( + u log t + O(u 2 )) + ( t)( + u log( t) + O(u 2 )) = + ( t log t + ( t) log( t) ) u + O(u 2 ). Hence (4.9) becomes ( + ( t log t + ( t) log( t) ) ) /u u + O(u 2 ) =. 3 In the limit for u 0, we obtain e t log t+( t) log( t) = 3. Solving t t ( t) t = 3, (4.0) we obtain t = Acknowledgement. The author would like to thank Gwoho Liu for computer assistance.

12 References [] J. Bourgain, A. Gamburd, P. Sarnak Affine linear sieve, expanders, and sum-product, Invent. Math., 79(3), , (200). [2] J. Bourgain, A. Kontorovich On the Local-Global Conjecture for integral Apollonian gaskets, Invent. Math., (204). arxiv: [3] J. Friedlander, H. Iwaniec Opera de cribro, Amer. Math. Soc., Providence, RI, (200). [4] H. Iwaniec and E. Kowalski Analytic number theory, Amer. Math. Soc., Providence, RI, (2004). [5] A. Kontorovich Levels of Distribution and the Affine Sieve Annales de la Faculté des Sci. Toulouse, To appear, (204) [6] E. Kowalski Sieve in Expansion, Séminaire Bourbaki, 63ème année, no 028, (200). [7] R. Peled, A. Sen, O. Zeitouni Double roots of random Littlewood polynomials, preprint, (204). arxiv: