Stephan Baier
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View article: Small solutions of ternary quadratic congruences with averaging over the moduli
Small solutions of ternary quadratic congruences with averaging over the moduli Open
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}…
View article: Diophantine approximation with sums of two squares
Diophantine approximation with sums of two squares Open
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…
View article: The large sieve for square moduli, revisited
The large sieve for square moduli, revisited Open
We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots.
View article: An uncountable subring of $\mathbb R$ with Hausdorff dimension zero
An uncountable subring of $\mathbb R$ with Hausdorff dimension zero Open
We construct a subring as mentioned in the title (hence this subring has Lebesgue measure zero).
View article: Small solutions of generic ternary quadratic congruences to general moduli
Small solutions of generic ternary quadratic congruences to general moduli Open
We study small non-trivial solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they consid…
View article: Diophantine Approximation with Piatetski-Shapiro Primes
Diophantine Approximation with Piatetski-Shapiro Primes Open
We prove that for every irrational number $α$, real number $β$, real number $c$ satisfying $1
View article: Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields
Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields Open
In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet $L$-functions to a fixed modulus over function fields. We apply the result to obtain upper bounds on moments of Dirichlet character sums over func…
View article: A lower bound for classical Kloosterman sums and an application
A lower bound for classical Kloosterman sums and an application Open
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 g…
View article: Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers
Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers Open
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\…
View article: Small Solutions of generic ternary quadratic congruences
Small Solutions of generic ternary quadratic congruences Open
We consider small solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. …
View article: Diophantine approximation with prime denominator in quadratic number fields under GRH
Diophantine approximation with prime denominator in quadratic number fields under GRH Open
Matomäki proved that if $α\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|α-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields un…
View article: Multiple exponential sums and their applications to quadratic congruences
Multiple exponential sums and their applications to quadratic congruences Open
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\g…
View article: A note on the distribution of prime ideals in real quadratic fields
A note on the distribution of prime ideals in real quadratic fields Open
In this note, we give a summary of the article ``The distribution of prime ideals of imaginary quadratic fields'' by G. Harman, A. Kumchev and P. A. Lewis and establish analogous results for real quadratic fields based on the same method.
View article: Diophantine approximation with prime denominator in real quadratic function fields
Diophantine approximation with prime denominator in real quadratic function fields Open
In the thirties of the last century, I.M. Vinogradov proved that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many auth…
View article: Large sieve inequalities with power moduli and Waring's problem
Large sieve inequalities with power moduli and Waring's problem Open
We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representation…
View article: Diophantine approximation with prime denominator in real quadratic function fields
Diophantine approximation with prime denominator in real quadratic function fields Open
In the thirties of the last century, I. M. Vinogradov proved that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently bee…
View article: A Bombieri–Vinogradov-type theorem for moduli with small radical
A Bombieri–Vinogradov-type theorem for moduli with small radical Open
In this article, we extend our recent work on a Bombieri–Vinogradov-type theorem for sparse sets of prime powers $p^N\leqslant x^{1/4-\varepsilon }$ with $p\leqslant (\log x)^C$ to sparse sets of moduli $s\leqslant x^{1/3-\varepsilon }$ wi…
View article: A Bombieri-Vinogradov-type theorem for moduli with small radical
A Bombieri-Vinogradov-type theorem for moduli with small radical Open
In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical rad$(s)\l…
View article: Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$
Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$ Open
Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for sol…
View article: Variance of primes in short residue classes for function fields
Variance of primes in short residue classes for function fields Open
Keating and Rudnick derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersec…
View article: Quantitative Oppenheim Conjecture for Quadratic Forms in 5 Variables over Function Fields
Quantitative Oppenheim Conjecture for Quadratic Forms in 5 Variables over Function Fields Open
We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.
View article: Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers
Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers Open
Let $p>5$ be a fixed prime and assume that $α_1,α_2,α_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $α_1x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|…
View article: Small Pythagorean triples modulo prime powers
Small Pythagorean triples modulo prime powers Open
Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.
View article: A Bombieri–Vinogradov-type theorem with prime power moduli
A Bombieri–Vinogradov-type theorem with prime power moduli Open
In 2020, Roger Baker proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal {S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l…
View article: Solving $p$-adic polynomial equations using Jarratt's Method
Solving $p$-adic polynomial equations using Jarratt's Method Open
We implement an iterative numerical method to solve polynomial equations $f(x)=0$ in the $p$-adic numbers, where $f(x) \in\mathbb{Z}_p[x]$. This method is a simplified $p$-adic analogue of Jarratt's method for finding roots of functions ov…
View article: On the Distribution of <i>αp</i> Modulo One in Quadratic Number Fields
On the Distribution of <i>αp</i> Modulo One in Quadratic Number Fields Open
We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and…
View article: Diophantine approximation with prime restriction in function fields
Diophantine approximation with prime restriction in function fields Open
In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||p…
View article: A Bombieri-Vinogradov-type theorem with prime power moduli
A Bombieri-Vinogradov-type theorem with prime power moduli Open
In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the…