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

collaborators

24 papers

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.CO2025

On exponential Freiman dimension

Jeck Lim, Akshat Mudgal, Cosmin Pohoata +1

The exponential Freiman dimension of a finite set $A \subset R^{m}$ , introduced by Green and Tao in 2006, represents the largest positive integer $d$ for which $A$ contain…

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.CO2024

Brunn-Minkowski type estimates for certain discrete sumsets

Albert Lopez Bruch, Yifan Jing, Akshat Mudgal

Let $d, k$ be natural numbers and let $L_{1}, \dots, L_{k} \in GL_{d} (Q)$ be linear transformations such that there are no non-trivial subspaces $U,…

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.CO2023

An Elekes-Rónyai theorem for sets with few products

Akshat Mudgal

Given $d, n \in N$ , we write a polynomial $F \in C [x_{1}, \dots, x_{n}]$ to be degenerate if there exist $P \in C [y_{1}, \dots, y_{n - 1}]$ and $m_j = x_1^{v_{j,1}}\…

math.CO2023

Kemperman's inequality and Freiman's lemma via few translates

Yifan Jing, Akshat Mudgal

Let $G$ be a connected compact group equipped with the normalised Haar measure $μ$ . Our first result shows that given $α, β > 0$ , there is a constant $c = c (α, β) > 0$ such that for any…

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.CA2022★ 1 cited

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

Brian Cook, Kevin Hughes, Zane Kun Li +3

We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely…

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.CO2021

New lower bounds for cardinalities of higher dimensional difference sets and sumsets

Akshat Mudgal

Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $R^{d}$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we sh…

math.CO2021★ 5 cited

Energy estimates in sum-product and convexity problems

Akshat Mudgal

We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set $A$ of integers may be written as $A = B \cup C$ , where $B$ and $C$ a…

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…

math.CO2020★ 3 cited

Difference sets in higher dimensions

Akshat Mudgal

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq R^{d}$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \…

math.CO2020

Sum-product estimates for diagonal matrices

Akshat Mudgal

Given $d \in N$ , we establish sum-product estimates for finite, non-empty subsets of $R^{d}$ . This is equivalent to a sum-product result for sets of diagonal matric…

math.CO2020

Arithmetic Combinatorics on Vinogradov systems

Akshat Mudgal

In this paper, we present a variant of the Balog-Szemerédi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product ph…

math.CO2019

Sums of linear transformations in higher dimensions

Akshat Mudgal

In this paper, we prove the following two results. Let $d$ be a natural number and $q, s$ be co-prime integers such that $1 < q s$ . Then there exists a constant $δ > 0$ depending only…

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

On exponential Freiman dimension

Counting 2×2 integer matrices with a given determinant

On commuting integer matrices

On commuting pairs in arbitrary sets of 2x2 matrices

Brunn-Minkowski type estimates for certain discrete sumsets

A Quadratic Vinogradov Mean Value Theorem in Finite Fields

An Elekes-Rónyai theorem for sets with few products

Kemperman's inequality and Freiman's lemma via few translates

Unbounded expansion of polynomials and products

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

Finding large additive and multiplicative Sidon sets in sets of integers

New lower bounds for cardinalities of higher dimensional difference sets and sumsets

Energy estimates in sum-product and convexity problems

Additive energies on spheres

Diameter free estimates for the quadratic Vinogradov mean value theorem

Difference sets in higher dimensions

Sum-product estimates for diagonal matrices

Arithmetic Combinatorics on Vinogradov systems

Sums of linear transformations in higher dimensions

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

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