arXiv subjects

Aishik Chattopadhyay

Publications and source records attributed to Aishik Chattopadhyay.

6 recordsLinked to original sources

arXiv2026

Burgess-type volume dependent bounds for character sums over $\mathbb{F}_{p^n}$

We establish a Burgess-type bound for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$. Let \[ B=\left\{\sum_{i=1}^{n}x_iω_i: N_i+1\le x_i\le N_i+H_i,1\le i\le n\right\}\subseteq\mathbb{F}_{p^n}, \] where $1\le H_i\le p$ for all $1\le i\le n$, and the side lengths satisfy $H_1\le H_2\le\cdots\le H_n.$ We prove that if the side lengths satisfy certain lower bounds in terms of the two largest side lengths, then a nontrivial cancellation occurs in the character sum over the boxes. This generalizes the work of Gabdullin \cite{GB} in dimensions $n=2,3$ to arbitrary dimension. This also generalizes the character sum estimate of Konyagin \cite{Kon} where each of the side lengths of the boxes are greater than $p^{1/4}$. The proof combines techniques from the geometry of numbers, multiplicative energy estimates, and Katz's bounds for multiplicative character sums.

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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.

math.NT↗

arXiv2026

Character sums over co-dimension one sub-lattices in finite field extensions

We obtain nontrivial bounds for multiplicative character sums over codimension-one sublattices of finite field extensions $\mathbb{F}_{p^d}$. This extends earlier results of Davenport--Lewis and Chang from full-dimensional settings to the codimension-one setting. As an application, we establish cancellation in character sums over binary cubic forms for intervals of length $p^{3/8+\varepsilon}$. Our approach combines Burgess-type amplification technique, represesentation of character sum using multiplicative energy and it's estimates to prove the result.

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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 $q$.

math.NT↗

arXiv2025

Small Solutions of generic ternary quadratic congruences

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 $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ as $q$ tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If $q$ is restricted to powers $p^m$ of a {\it fixed} prime $p$ and $m$ tends to infinity, we obtain a slight improvement of this result using the theory of $p$-adic exponent pairs, as developed by Milićević, replacing the exponent $11/24$ above by $11/25$. Under the Lindelöf hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$.

math.NT↗

arXiv2024

Small solutions of generic ternary quadratic congruences to general moduli

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 considered the case of prime power moduli $q$. Above, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients.

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