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The Quarterly Journal of Mathematics
, Volume Advance Article (2) – Nov 22, 2017

13 pages

/lp/ou_press/linear-equations-with-rational-fractions-of-bounded-height-and-PbueYdp6Ev

- Publisher
- Oxford University Press
- Copyright
- © The Author 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hax049
- Publisher site
- See Article on Publisher Site

Abstract We obtain a tight, up to a logarithmic factor, upper bound on the number of solutions to the equation ∑j=1najsjrj=a0,with variables r1,…,rn in an arbitrary box at the origin and variables s1,…,sn in an essentially arbitrary translation of this box. We apply this result to get an upper bound on the number of stochastic matrices with rational entries of bounded height. 1. Introduction 1.1. Motivation We recall that a stochastic matrix is a square n×n matrix A=(αi,j)i,j=1n with non-negative entries and such that ∑j=1nαi,j=1,i=1,…,n. (1.1) Furthermore, as usual, for a rational number α we define its height h(α) as max{∣s∣,r}, where the integers r,s∈Z are uniquely defined by the conditions α=s/r, gcd(r,s)=1, r≥1. We now define Sn(H) as the number of stochastic n×n matrices with rational entries of height at most H, that is, with entries from the set F(H)={α∈Q:α≥0,h(α)≤H}.In particular, F(H)∩[0,1] is the classical set of Farey fractions of order H. The question of estimating Sn(H) seems to be quite natural; however, it has never been addressed in the literature. Since the conditions (1.1) are independent, we clearly have Sn(H)=Ln(H)n, (1.2)where Ln(H) is the number of solutions to the linear equation ∑j=1nαj=1,αj∈F(H),j=1,…,n,which we also write as ∑j=1nsjrj=1H≥rj≥sj≥0,gcd(rj,sj)=1,j=1,…,n. (1.3) We remark that each solution to (1.3) leads to an integer solution to ∑j=1nmjkj=0,∣kj∣,∣mj∣≤K, (1.4)with K=2H. Indeed, it is enough to set (kj,mj)=(rj,sj), j=1,…,n−1, and (kn,mn)=(rn,sn−rn). Furthermore, distinct solutions to (1.3) yields distinct solutions to (1.4). We now recall that for n=3, the result of Blomer et al. [2] gives an asymptotic formula K3Q(logK)+O(K3−δ) for the number of solutions to (1.4), where Q∈Q[X] is a polynomial of degree 4 and δ>0 is some absolute constant. This immediately implies the bound L3(H)=O(H3(logH)4).It is also noted in [2] that the same method is likely to work for any n, it may also work for Equation (1.3) directly and give an asymptotic formula for Ln(H). The more elementary approach of Blomer and Brüdern [1] can probably be used to get an upper bound on Ln(H) of the right order of magnitude. However, working out the above approaches from [1, 2] in full detail may require significant efforts. Here we suggest an alternative way to estimate Ln(H) via modular reduction modulo an appropriate prime and bounds on some double exponential sums with rational fractions. Although the bound obtained via this approach does not reach the same strength as the hypothetical results that can be derived via the methods of Blomer et al. [2] or Blomer and Brüdern [1], it is weaker only by a power of a logarithm. Furthermore, the suggested method here seems to be more robust and also applies to more general equations, see (1.7) below. Besides, it also works for variables in distinct intervals and not necessary at the origin. However, it is unable to produce asymptotic formulas. Clearly, to estimate Sn(H), one can use general bounds for the number of integral points on hypersurfaces, see, for example, [7] and references therein, however, they do not reach the strength of our results. 1.2. Main results We derive the bounds Ln(H) and thus on Sn(H) from a bound on the number of solutions of a much more general linear equation. Namely, we consider two boxes of the form B0=[1,H1]×⋯×[1,Hn] (1.5)and B=[B1+1,B1+H1]×⋯×[Bn+1,Bn+Hn], (1.6)with arbitrary integers Bj and positive integers Hj, i=1,…,n. We note that the boxes B0 and B are of the same dimensions but B0 is positioned at the origin, while B can be at an arbitrary location in Rn. Namely, for a vector a=(a0,a1,…,an)∈Zn+1, we use Nn(a;B0,B) to denote the number of solutions to the equation ∑j=1najsjrj=a0,(r1,…,rn)∈B0,(s1,…,sn)∈B. (1.7)We remark that we have dropped the condition of co-primality of the variables, which only increases the number of solutions, but does not affect our main results. Throughout the paper, any implied constants in the symbols O, ≪ and ≫ may depend on the real parameter ε>0 and the integer parameter n≥1. We recall that the notations U=O(V), U≪V and V≫U are all equivalent to the statement that the inequality ∣U∣≤cV holds with some constant c>0. Theorem 1.1 Let Band B0be two boxes of the form (1.5) and (1.6), respectively, with maxj=1,…,nHj=H. For any vector a=(a0,a1,…,an)∈Zn+1with 1≤∣ai∣≤exp(HO(1)), i=1,…,n, we have Nn(a;B0,B)≤H1,…,Hn(logH)2n−1+o(1). We remark that in Theorem 1.1, there is no restriction on the size or non-vanishing of a0 on the right hand size of (1.7). Corollary 1.2 We have Hn2≪Sn(H)≤Hn2(logH)n(2n−1)+o(1). We derive these results via modular reduction of Equation (1.7) modulo an appropriately chosen prime p. In turn, to estimate the number of solutions of the corresponding congruence, we use a new bound of double exponential sums, which slightly improves [9, Lemma 3]. We present this result in a larger generality than is needed for proving Theorem 1.1 as we believe it may be of independent interest. Furthermore, since there is a gap between upper and lower bounds of Corollary 1.2, it is natural to do some numerical experiments and try to understand the asymptotic behaviour of Sn(H) as H→∞. Thus, it is interesting to design an efficient algorithm to compute Sn(H). Since by (1.2), it is enough to compute Ln(H) we see that one can do this via the following naive algorithm: for each choice of n−1 positive integers ri≤H, i=1,…,n−1, then each choice of n−1 non-negative integers si<ri, with gcd(si,ri)=1, i=1,…,n−1, check whether 1−s1/r1−⋯−sn−1/rn−1∈F(H).This obviously gives an algorithm of complexity H2n−2+o(1). We now show that one can compute Sn(H) faster. Theorem 1.3 There is a deterministic algorithm to compute Sn(H)of complexity H3n/2−1+o(1). 2. Preliminaries 2.1. Background on totients Let φ(r) denote the Euler function. We need the following well-known consequence of the sieve of Eratosthenes. Lemma 2.1 For any integers r≥ℓ≥1, ∑s=1gcd(s,r)=1ℓ1=φ(r)rℓ+O(ro(1)). Proof For an integer d≥1, we use μ(d) to denote the Möbius function. We recall that μ(1)=1, μ(d)=0 if d≥2 is not square-free, and μ(d)=(−1)ω(d) otherwise, where ω(d) is the number of prime divisors of d. Now, using the Möbius function μ(d) over the divisors of r to detect the co-primality condition and interchanging the order of summation, we obtain ∑s=1gcd(s,r)=1ℓ1=∑d∣rμ(d)⌊ℓd⌋=ℓ∑d∣rμ(d)d+O(∑d∣r∣μ(d)∣).We now use the well-known identity ∑d∣rμ(d)d=φ(r)r,see [4, Section 16.3], and also that ∑d∣r∣μ(d)∣=2ω(r),see [4, Theorem 264], which yield ∑s=1gcd(s,r)=1ℓ1=φ(r)rℓ+O(2ω(r)).Since obviously ω(r)!≤r, the result now follows immediately.□ One can certainly obtain a much more precise version of the following statement, which we present in a rather crude form that is, however, sufficient for our applications. Lemma 2.2 For any n≥1we have ∑r=1Hφ(r)n−1≫Hn. Proof By the Hölder inequality, (∑r=1Hφ(r))n−1≤Hn−2∑r=1Hφ(r)n−1.Using the classical asymptotic formula ∑r=1Hφ(r)=(3π2+o(1))H2,see [4, Theorem 330], we conclude the proof.□ 2.2. Background on divisors For an integer m≥1, we use τ(m) to denote the divisor function τ(m)= #{d∈Z:d≥1,d∣m}and Δ(m) to denote the Hooley function Δ(m)=maxu≥0#{d∈Z:u<d≤eu,d∣m}. We need upper bounds on the average values of τ(m) and Δ(m), where m runs through terms of arithmetic progressions indexed by prime numbers. We derive these bounds from very general results of Nair and Tenenbaum [8]. We start with the τ(m). Lemma 2.3 For any fixed real ε>0and integer n≥1, for positive integers a, mand M≥max{a,mε}, we have ∑M≤p≤2Mpprimeτ(a+pm)n≪M(logM)2n−2. Proof We apply [8, Theorem 3] (taken with the polynomials Q(X)=mX+a and x=y=M), and derive ∑M≤p≤2Mpprimeτ(a+pm)n≪MlogM∏p≤Mpprime(1−1p)∑m≤Mτ(m)nm,where the implied constant depends only on ε and n. Using the Mertens formula for the product over primes, and also the classical bound of Mardjanichvili [6] ∑m≤Mτ(m)n≪M(logM)2n−1 (2.1)(combined with partial summation), we easily derive the result.□ For Δ(m) we obtain a slightly stronger bound. Lemma 2.4 For any fixed real ε>0and integer n≥1, for positive integers a, mand M≥max{a,mε}, we have ∑M≤p≤2MpprimeΔ(a+pm)n≪M(logM)2n−n−2+o(1). Proof As in the proof of Lemma 2.3, by [8, Theorem 3], we have ∑M≤p≤2MpprimeΔ(a+pm)n≪MlogM∏p≤Mpprime(1−1p)∑m≤MΔ(m)nm,where the implied constant depends only on ε and n. Now, using [10, Lemma 2.2] instead of (2.1), we conclude the proof.□ We remark that using the full power of [8, Theorem 3] and [10, Lemma 2.2], one can replace o(1) in the power of logM in Lemma 2.4 by a more precise and explicit function of M. 2.3. Product and least common multiples of several integers We need the following result of Karatsuba [5]. For an integer n≥1 and real R1,…,Rn, let Jn(R1,…,Rn) denote the number of solutions to the congruence r1,…,rn≡0(modlcm[r12,…,rn2]) (2.2)in positive integers rj≤Rj, j=1,…,n. Lemma 2.5 For any real R1,…,Rn≥2, we have Jn(R1,…,Rn)≪R1/2(logR)n2,where R=R1,…,Rn. 2.4. Exponential sums with ratios For a prime p, we denote ep(z)=exp(2πiz/p). Clearly for any p>u,v≥1, the expression ep(av/u) is correctly defined (as ep(aw) for w≡v/u(modp)). The following result is a variation of [9, Lemma 3], where the additional averaging over primes allows us to replace Qo(1) with a power of logQ. Lemma 2.6 Let Q>U,V≥1be arbitrary integers and let C⊆[0,U]×[0,V]be an arbitrary convex domain. Then, uniformly over the integers a∈[1,2Q], we have ∑Q≤p≤2Qpprimegcd(a,p)=1∣∑(u,v)∈Cep(av/u)∣n≪(U+V)nQ(logQ)2n−2+o(1). Proof Since C is convex, for each v, there are integers V≥Wu>Vu≥0 such that ∑(u,v)∈Cep(av/u)=∑u=1U∑v=Vu+1Wuep(av/u).Following the proof of [9, Lemma 3], we define I=⌊log(2Q/V)⌋andJ=⌊log(2Q)⌋. Furthermore, for a rational number α=v/u with gcd(u,p)=1, we denote by ρp(α) the unique integer w with w≡v/u(modp) and ∣w∣<p/2. Then [9, Equation (1)] implies ∑(u,v)∈Cep(av/u)≪VRp+Q∑j=IJTj,pe−j,where Rp=#{u:1≤u≤U,∣ρp(a/u)∣<eI},Tj,p=#{u:1≤u≤U,ej≤∣ρp(a/u)∣<ej+1}. Thus, using the Hölder inequality twice, we obtain ∣∑(u,v)∈Cep(av/u)∣n≪VnRpn+Qn(∑j=IJTj,pe−j)n≪VnRpn+Qn(logQ)n−1∑j=IJTj,pne−jn.Hence ∑Q≤p≤2Qpprimegcd(a,p)=1∣∑(u,v)∈Cep(av/u)∣n≪VnR+Qn(logQ)n−1∑j=IJTje−jn, (2.3)where R=∑Q≤p≤2Qpprimegcd(a,p)=1RpnandTj=∑Q≤p≤2Qpprimegcd(a,p)=1Tj,pn. As in [9], we note that if ej≤∣ρp(a/u)∣<ej+1, then uz≡a(modp) for some integer z with ej<∣z∣<ej+1. Thus uz=a+pk for some integer k with ∣k∣≤Kj, where Kj=⌊ej+1U/Q⌋+1. Therefore, recalling the definitions of the τ(m) and Δ(m), we conclude Rp≤∑∣k∣≤KI−1τ(∣a+pk∣)andTj,p≤∑∣k∣≤KjΔ(∣a+pk∣).Now using the Hölder inequality, changing the order of summation and applying Lemma 2.3 (which applies as Q≫KJ≥∣k∣), we derive R≤∑Q≤p≤2Qpprime(∑∣k∣≤KIτ(∣a+pk∣))n≤KIn−1∑∣k∣≤KI∑Q≤p≤2Qpprimeτ(∣a+pk∣)n≪KInQ(logQ)2n−2.Similarly, applying Lemma 2.4, we see that Tj≤∑Q≤p≤2Qpprime(∑∣k∣≤KjΔ(∣a+pk∣))n≤Kjn−1∑∣k∣≤Kj∑Q≤p≤2QpprimeΔ(∣a+pk∣)n≪KjnQ(logQ)2n−n−2+o(1),for j=I,…,J. Substituting this bound in (2.3) yields ∑Q≤p≤2Qpprimegcd(a,p)=1∣∑(u,v)∈Cep(av/u)∣n≪VnKInQ(logQ)2n−2+Qn+1(logQ)2n−3+o(1)∑j=IJe−jnKjn. (2.4) We now have KIn≪eInUnQ−n+1≪UnV−n+1and also ∑j=IJe−jnKjn≪∑j=IJ(ejnUnQ−n+1)e−jn≪JUn/Qn+e−In≪UnQ−nlogQ+VnQ−n.Combining the above bounds with (2.4), after simple calculations, we obtain the desired result.□ 3. Proofs of Main Results 3.1. Proof of Theorem 1.1 We note that since ∣a1,…,an∣=exp(HO(1)), this product has at most HO(1) prime divisors. Hence, there is a constant C>0 such that for Q=⌈HC⌉, there is a set of at least #≥0.5Q/logQ (3.1)primes p∈[Q,2Q] that are relatively prime with a1,…,an. We also assume that C>n. In particular, HC≫Q>Hn. (3.2) Let Mn(a;p,B0,B) be the number of solutions to the congruence ∑j=1najsjrj≡a0(modp),(r1,…,rn)∈B0,(s1,…,sn)∈B. Using the orthogonality of exponential functions, we write Mn(a;p,B0,B)=∑∑(r1,…,rn)∈B0(s1,…,sn)∈B1p∑λ=0p−1ep(λ(∑j=1najsjrj−a0)).Changing the order of summation gives the identity Mn(a;p,B0,B)=1p∑λ=0p−1ep(−λa0)∏j=1n∑rj=1Hj∑sj=Bj+1Bj+Hjep(λajsj/rj).Now, the contribution from λ=0 gives the main term (H1,…,Hn)2/p. Extending the summation over λ to all positive integers λ≤2Q with gcd(λ,p)=1, for every p∈, we obtain Mn(a;p,B0,B)≤(H1,…,Hn)2Q+1Q∑λ=1gcd(λ,p)=12Q∏j=1n∣∑rj=1Hj∑sj=Bj+1Bj+Hjep(λajsj/rj)∣.Hence, summing over all p∈ and denoting W(λ)=∑p∈gcd(λ,p)=1∏j=1n∣∑rj=1Hj∑sj=Bj+1Bj+Hjep(λajsj/rj)∣,we see that ∑p∈Mn(a;p,B0,B)≤#(H1,…,Hn)2Q+1Q∑λ=1gcd(λ,p)=12QW(λ). (3.3)Using the Hölder inequality, we obtain W(λ)n≤∏j=1n∑p∈gcd(λ,p)=1∣∑rj=1Hj∑sj=Bj+1Bj+Hjep(λajsj/rj)∣n.We now invoke Lemma 2.6 and see that W(λ)≪H1,…,HnQ(logQ)2n−2+o(1),which together with (3.3) implies ∑p∈Mn(a;p,B0,B)≪#(H1,…,Hn)2Q+H1,…,HnQ(logQ)2n−2+o(1).Hence, by (3.1), there is a prime p∈ with Mn(a;p,B0,B)≪(H1,…,Hn)2Q+H1,…,Hn(logQ)2n−1+o(1).Recalling (3.2), we see that the second term dominates and logQ can be replaced with logH. Therefore, Mn(a;p,B0,B)≪H1,…,Hn(logH)2n−1+o(1).Using the trivial bound Nn(a;B0,B)≤Mn(a;p,B0,B), we conclude the proof. 3.2. Proof of Corollary 1.2 The upper bound is immediate from Theorem 1.1 and Equation (1.2) (we note that for the typographical simplicity the values sj=0 are excluded in Theorem 1.1, but a simple inductive argument allows us to include them). To see the lower bound we note that by Lemma 2.1 for any positive integer r≤H and ℓ=⌊r/(n−1)⌋ and we can choose positive integers s1,…,sn−1≤ℓ with gcd(s1,…,sn−1,r)=1 (3.4)in ((1+o(1))φ(r)rℓ)n−1≫φ(r)n−1possible ways. After this, we set sn=r−s1−⋯−sn−1. Clearly, by the co-primality condition (3.4), the vectors of rational numbers (s1/r,…,sn/r) obtained via the above construction are pairwise distinct. Hence, Ln(H)≫∑r=1Hφ(r)n−1.Applying Lemma 2.2, we obtain Ln(H)≫Hn and the result follows from (1.2). 3.3. Proof of Theorem 1.3 Clearing the denominators, we transform (1.3), into the following equation: ∑j=1nsjr1,…,rnrj=r1,…,rn.Since for every j=1,…,n, we have gcd(rj,sj)=1, we see the divisibility rj∣r1,…,rnrjor rj2∣r1,…,rn, which implies the congruence (2.2) for every solution to (1.3). Using standard arithmetic algorithms for computing the greatest common divisor and the least common multiple, (see [3]), we see that we can enumerate all vectors of positive integers (r1,…,rn) for which (1.3) has a solution for some s1,…,sn in time O(Hn+o(1)). By Lemma 2.5, the resulting list contains O(Hn/2+o(1)) vectors. Now for each (r1,…,rn), we choose 0≤sj<rj, j=1,…,n−1 in O(Hn−1) ways, define sn by Equation (1.3) and check whether other conditions in (1.3). This leads to the desired algorithm. 4. Comments We remark that estimating Ln(H) by the number of solutions to an equation of the type (1.7) (that is, without the co-primality condition) can lead to additional logarithmic losses. This effect has been mentioned in [2] and can also be easily seen for n=2. We recall that a square n×n matrix A is called doubly stochastic if both A and the transposed matrix AT are stochastic. We now define Sn(H) as the number of doubly stochastic n×n matrices with rational entries from F(H). We have the following trivial bounds: Hn2−2n+2≪Sn(H)≤Hn2−n(logH)(n−1)(2n−1)+o(1). (4.1)Indeed the upper bound in (4.1) follows immediately from Theorem 1.1 and the observation that if the top n−1 rows of a doubly stochastic matrix are fixed then the last row is uniquely defined. To get a lower bound on Sn(H), we fix a positive integer r and for each i=1,…,n−1 we choose the elements αi,j=si,j/r, i=1,…,n−1, j=1,…,n, of the first n−1 rows as in the proof of Corollary 1.2 (with respect to the same r). After this we also define αn,j=1−∑i=1n−1αi,j,j=1,…,n.It only remains to note that ∑j=1nαn,j=n−∑j=1n∑i=1n−1αi,j=n−∑i=1n−1∑j=1nαi,j=1.Now, simple counting yields the lower bound in (4.1). Acknowledgement The author is grateful to Valentin Blomer and Tim Browning for very useful discussions. Funding This work was supported in part by ARC Grant DP140100118. References 1 V. Blomer and J. Brüdern, The Density of Rational Points on a Certain Threefold, Contributions in Analytic and Algebraic Number Theory , Springer, Berlin, 2012, 1– 15. Google Scholar CrossRef Search ADS 2 V. Blomer, J. Brüdern and P. Salberger, On a certain senary cubic form, Proc. Lond. Math. Soc. 108 ( 2014), 911– 964. Google Scholar CrossRef Search ADS 3 J. von zur Gathen and J. Gerhard, Modern Computer Algebra , Cambridge University Press, Cambridge, 2013. Google Scholar CrossRef Search ADS 4 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers , Oxford University Press, Oxford, 1979. 5 A. A. Karatsuba, Analogues of Kloosterman sums, Izvestiya Mathematics (Translated from Izvestiya RAN) 59 ( 1995), 971– 981. Google Scholar CrossRef Search ADS 6 C. Mardjanichvili, Estimation d’une somme arithmétique, Dokl. Akad. Nauk. SSSR. 22 ( 1939), 387– 389. 7 O. Marmon, The density of integral points on hypersurfaces of degree at least four, Acta Arith. 141 ( 2010), 211– 240. Google Scholar CrossRef Search ADS 8 M. Nair and G. Tenenbaum, Short sums of certain arithmetic functions, Acta Math. 180 ( 1998), 119– 144. Google Scholar CrossRef Search ADS 9 I. E. Shparlinski, Exponential sums with Farey fractions, Bull. Polish Acad. Sci. Math. 57 ( 2009), 101– 107. Google Scholar CrossRef Search ADS 10 G. Tenenbaum, Sur une question d’Erdös et Schinzel, II, Invent. Math. 99 ( 1990), 215– 224. Google Scholar CrossRef Search ADS © The Author 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Quarterly Journal of Mathematics – Oxford University Press

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