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Stephan Baier

Publications and source records attributed to Stephan Baier.

At least 19 recordsLinked to original sources
arXiv2026

The large sieve for square moduli under Hooley's hypothesis $R^*$

Let $S(Q,M,N,(a_n)):=\sum_{q\le Q}\sum_{(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. At the critical point $N=Q^3$ the best known unconditional bound, due to Baier and Zhao (2008), is $S\ll Q^{1/2+\varepsilon}N\sum |a_n|^2$, against the conjectured $Q^{\varepsilon}N\sum|a_n|^2$, and the exponent $\tfrac12$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^*$ for short Salié sums -- square-root cancellation for $\sum_{x_1<n\le x_2}\big(\tfrac nc\big)e_c(a\bar n+bn)$ over arbitrary subintervals of a period -- one has $S\ll Q^{1/2-1/134+\varepsilon}N\sum|a_n|^2$ at $N=Q^3$. The key estimate is a bound for the number $P(α)$ of fractions $a/q^2$, $q\le Q$, within $Q^{-3}$ of a point $α$ near $b/r$: we show $P(b/r+z)\ll(Q^{2/3}r^{-1/3}+Q^{1/4})Q^\varepsilon$ for every modulus $Q^{1/2+\varepsilon}\le r\le Q^{3/2}$, improving the bound $Q^{9/16}r^{-1/8}$ obtained by Baier (2026) for $r=p,p^2$ only, and reaching every modulus. The proof rests on a single observation: a sum of modular square roots $\sum_{n\in J}e_r(a\sqrt{jn})$ over an interval $J$ is, after completion and an exact evaluation of quadratic Gauss sums at every modulus, $r^{-1/2}$ times a Salié sum of length $r/|J|$. Hypothesis $R^*$ therefore yields square-root cancellation for these sums directly, at every modulus, without Weyl differencing; the saving over the trivial bound is the square of what the Weyl-differencing route gives. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. The paper was prepared in collaboration with Claude (Anthropic); Section 1.9 sets out what each of us contributed.

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arXiv2026

Character sums to prime power moduli evaluated at binary quadratic forms

We establish estimates for short character sums to prime power moduli evaluated at binary quadratic forms. This complements estimates established by Heath-Brown for such character sums to squarefree moduli. Our approach uses $p$-adic analysis. More precisely, we use tools from the $p$-adic theory of exponential sums, as initiated by Milićević.

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arXiv2026

On bilinear sums with modular square roots and applications II

In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make further partial progress on the large sieve for square moduli.

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arXiv2026

On bilinear sums with modular square roots and applications III

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.

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arXiv2026

Partial progress towards the large sieve for square moduli

This paper condenses and expands upon our recent contributions arXiv:2603.00768 and arXiv:2603.25814, which belong together, producing a single streamlined manuscript. We have added consequences of an analogue of Hooley's hypothesis R$^{\ast}$ for short Salié sums. The motivation for these contributions comes from the large sieve for square moduli, which presents a specialized but compelling research object. The first part arXiv:2601.15448 of this series used geometry of numbers and established new conditional results on the large sieve for square moduli under reasonable hypotheses on additive energies of modular square roots. In contrast, we here explore how much can be achieved unconditionally by using purely Fourier analytic techniques. While our progress is limited, it is worthwhile to exhaust these classical techniques first before moving to other ideas. This may be of relevance for a potential future resolution of this problem (particularly in the function field setting). The central problem is to estimate the number of fractions with square moduli in short intervals.

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arXiv2026

Diophantine approximation with sums of two squares II

Recently, the authors showed that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||αn|| 0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $γ<3/7-\varepsilon$. In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$, where we confine ourselves to the particular case of sums of two squares.

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arXiv2026

On certain bilinear sums with modular square roots and applications

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.

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arXiv2026

Diophantine approximation with primes from short intervals

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $α$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $α$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||pα||<δ$ is asymptotically equal to $2δY/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le δ\le 1/2$.

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arXiv2026

Diophantine approximation with sums of two squares

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||αn|| 0$. This is an easy consequence of a result by Cook on small fractional parts of diagonal quadratic forms. Secondly, we give a quantitative version with a lower bound of this result when the exponent $1/2-\varepsilon$ is replaced by any fixed $γ<3/7$. To this end, we use the Voronoi summation formula and a bound for bilinear forms with Kloosterman sums to fixed moduli by Kerr, Shparlinski, Wu and Xi.

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arXiv2026

Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $λ_1x_1^2+\cdots +λ_nx_n^2\equiv λ_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $λ_1,...,λ_{n+1}$ are integer coefficients such that $(λ_1\cdots λ_{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $λ_1,...,λ_n>0$ and $(λ_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $λ_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $λ_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag.

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arXiv2026

Small solutions of ternary quadratic congruences with averaging over the moduli

In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed $α_2\in \mathbb{N}$ coprime to $q$, the congruence \[ x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q} \] has a solution of $(x_1,x_2,x_3)\in \mathbb{Z}^3$ with $x_3$ coprime to $q$ of height $\max\{|x_1|,|x_2|,|x_3|\}\le q^{11/24+\varepsilon}$ for, in a sense, almost all $α_3$, where $α_3$ runs over the reduced residue classes modulo $q$. Here it was of significance that $11/24<1/2$, so we broke a natural barrier. In this paper, we average the moduli $q$ in addition, establishing the existence of a solution of height $\le Q^{3/8+\varepsilon}α_2^{\varepsilon}$ for almost all pairs $(q,α_3)$, with $Q$ large enough, $Q<q\le 2Q$, $q$ coprime to $2α_2$ and $α_3$ running over the reduced residue classes modulo $q$.

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arXiv2026

A lower bound for classical Kloosterman sums and an application

We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$.

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arXiv2025

Multiple exponential sums and their applications to quadratic congruences

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistribution result for these solutions.

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arXiv2025

Diophantine Approximation with Piatetski-Shapiro Primes

We prove that for every irrational number $α$, real number $β$, real number $c$ satisfying $1<c<9/8$ and positive real number $θ$ satisfying $θ<(9/c-8)/10$, there exist infinitely many primes of the form $p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||αp||<p^{-θ}$.

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