Stephan Baier

Solutions of $x_{1}^{2} + x_{2}^{2} - x_{3}^{2} = D$ with small $x_{3}$ : a spectral approach

Stephan Baier

We count the integral solutions of $x_{1}^{2} + x_{2}^{2} - x_{3}^{2} = D$ weighted by $Φ (x_{3} / M)$ , for $M$ smaller than $D$ . Such solutions correspond to binary quadratic forms of discriminant…

An improvement of the large sieve for square moduli under Hooley's hypothesis $R^{*}$ for short Salié sums

Stephan Baier

Let $S (Q, M, N, (a_{n})) = \sum_{q \leq Q} \sum_{a mod q^{2}, (a, q) = 1} ∣ \sum_{M < n \leq M + N} a_{n} e (an / q^{2}) ∣^{2}$ be Zhao's large sieve sum with square moduli. Zhao conjectured that $S\ll(QN)^ε(Q^3+N)\…

Partial progress towards the large sieve for square moduli

Stephan Baier

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 adde…

On bilinear sums with modular square roots and applications III

Stephan Baier

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 an…

On bilinear sums with modular square roots and applications II

Stephan Baier

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 ra…

On certain bilinear sums with modular square roots and applications

Stephan Baier

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 expone…

Diophantine approximation with primes from short intervals

Stephan Baier, Sayantan Roy

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:…

Small solutions of ternary quadratic congruences with averaging over the moduli

Stephan Baier, Aishik Chattopadhyay

In a recent paper, we proved that for any large enough odd modulus $q \in N$ and fixed $α_{2} \in N$ coprime to $q$ , the congruence \[ x_1^2+α_2x_2^2+α_3x_3^2\equiv 0…

A Bombieri-Vinogradov theorem for sectors in real quadratic number fields

Stephan Baier, Esrafil Ali Molla

We establish a Bombieri-Vinogradov theorem for sectors in real quadratic number fields.

Diophantine approximation with sums of two squares II

Stephan Baier, Habibur Rahaman

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 for…

Character sums to prime power moduli evaluated at binary quadratic forms

Stephan Baier, Aishik Chattopadhyay

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…

Diophantine approximation with sums of two squares

Stephan Baier, Habibur Rahaman

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…

The large sieve for square moduli, revisited

Stephan Baier

We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots.

An uncountable subring of $R$ with Hausdorff dimension zero

Stephan Baier, Shameek Paul

We construct a subring as mentioned in the title (hence this subring has Lebesgue measure zero).

Small solutions of generic ternary quadratic congruences to general moduli

Stephan Baier, Aishik Chattopadhyay

We study small non-trivial solutions of quadratic congruences of the form $x_{1}^{2} + α_{2} x_{2}^{2} + α_{3} x_{3}^{2} \equiv 0 mod q$ , with $q$ being an odd natural number, in an average sense. Thi…

Diophantine Approximation with Piatetski-Shapiro Primes

Stephan Baier, Habibur Rahaman

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…

A lower bound for classical Kloosterman sums and an application

Stephan Baier, Jishu Das, Jewel Mahajan

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 explic…

Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

Stephan Baier, Arkaprava Bhandari, Anup Haldar

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $λ_{1} x_{1}^{2} + \dots + λ_{n} x_{n}^{2} \equiv λ_{n + 1} mod p^{m}$ , where $p$ is a fixed odd prime,…