Akshat Mudgal

activity

20192026

most citedEnergy estimates in sum-product and convexity problems

5 citations · 9 across the 22 of their papers we have counts for

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13 papers · 1 filter

math.NT2026

On an asymmetric additive energy inequality

Akshat Mudgal

Let $d \geq 1$ be an integer, $G$ be an abelian group and $ν, w_{1}, \dots, w_{2 d} : G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive e…

math.NT2026

Counting solutions to the quadratic determinant equation

Jonathan Chapman, Akshat Mudgal

Given $h, N \in N$ satisfying $1 ⩽ h ⩽ N^{2}$ , we prove an asymptotic formula for the number of solutions to the equation $x_{1} x_{2} - x_{3} x_{4} = h$ with $x_1,…

math.NT2026

A structure theorem for sets with doubling $4 + δ$

Yifan Jing, Akshat Mudgal

We prove a structural result for sets of integers with doubling at most $4 + δ$ , with $δ > 0$ sufficiently small. This generalises earlier work of Eberhard--Green--Manners which deal…

math.NT2026

Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture

Joseph Harrison, Akshat Mudgal, Harry Schmidt

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this…

math.NT2025

Generalised Fermat equations in dense variables over finite fields and rings

Sam Chow, Zi Li Lim, Akshat Mudgal

Let $A$ be a sufficiently dense subset of a finite field $F_{q}$ or a finite, cyclic ring $Z / N Z$ . Assuming that $q$ and $N$ have no small prime divisors, we…

math.NT2025

Counting $2 \times 2$ integer matrices with a given determinant

Jonathan Chapman, Akshat Mudgal

Given positive integers $h, N$ satisfying $1 ⩽ h ⩽ 2 N^{2}$ , we define $T (h, N)$ to be the number of $2 \times 2$ integer matrices with determinant equal to $h$ whose en…

math.NT2025

On commuting integer matrices

Jonathan Chapman, Akshat Mudgal

Given $d, N \in N$ , we define $C_{d} (N)$ to be the number of pairs of $d \times d$ matrices $A, B$ with entries in $[- N, N] \cap Z$ such that $A B = B A$ . We…

math.NT2024

On commuting pairs in arbitrary sets of 2x2 matrices

Akshat Mudgal

Let $Mat_{2} (R)$ be the set of $2 \times 2$ matrices with real entries. For any $ε > 0$ and any finitely--supported probability measure $μ$ on $\textrm{Mat}…

math.NT2023

A Quadratic Vinogradov Mean Value Theorem in Finite Fields

Samuel Mansfield, Akshat Mudgal

Let $p$ be a prime, let $s \geq 3$ be a natural number and let $A \subseteq F_{p}$ be a non-empty set satisfying $∣ A ∣ ≪ p^{1/2}$ . Denoting $J_{s} (A)$ to be the number of sol…

math.NT2023

Unbounded expansion of polynomials and products

Akshat Mudgal

Given $d, s \in N$ , a finite set $A \subseteq Z$ and polynomials $φ_{1}, \dots, φ_{s} \in Z [x]$ such that $1 \leq d e g φ_{i} \leq d$ for every $1 \leq i \leq s…

math.NT2022

Finding large additive and multiplicative Sidon sets in sets of integers

Yifan Jing, Akshat Mudgal

Given $h, g \in N$ , we write a set $X \subset Z$ to be a $B_{h}^{+} [g]$ set if for any $n \in Z$ , the number of solutions to the additive equation $n = x_…

math.NT2021

Additive energies on spheres

Akshat Mudgal

In this paper, we study additive properties of finite sets of lattice points on spheres in $3$ and $4$ dimensions. Thus, given $d, m \in N$ , let $A$ be a set of lattice poi…

math.NT2020

Diameter free estimates for the quadratic Vinogradov mean value theorem

Akshat Mudgal

Let $s \geq 3$ be a natural number, let $ψ (x)$ be a polynomial with real coefficients and degree $d \geq 2$ , and let $A$ be some large, non-empty, finite subset of real numbers. We…

On an asymmetric additive energy inequality

Counting solutions to the quadratic determinant equation

A structure theorem for sets with doubling 4+δ

Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture

Generalised Fermat equations in dense variables over finite fields and rings

Counting 2×2 integer matrices with a given determinant

On commuting integer matrices

On commuting pairs in arbitrary sets of 2x2 matrices

A Quadratic Vinogradov Mean Value Theorem in Finite Fields

Unbounded expansion of polynomials and products

Finding large additive and multiplicative Sidon sets in sets of integers

Additive energies on spheres

Diameter free estimates for the quadratic Vinogradov mean value theorem

A structure theorem for sets with doubling $4 + δ$

Counting $2 \times 2$ integer matrices with a given determinant