arXiv subjects

Akshat Mudgal

Publications and source records attributed to Akshat Mudgal.

24 records · Page 2Linked to original sources

arXiv2022

Additive energies on spheres

In this paper, we study additive properties of finite sets of lattice points on spheres in $3$ and $4$ dimensions. Thus, given $d,m \in \mathbb{N}$, let $A$ be a set of lattice points $(x_1, \dots, x_d) \in \mathbb{Z}^d$ satisfying $x_1^2 + \dots + x_{d}^2 = m$. When $d=4$, we prove threshold breaking bounds for the additive energy of $A$, that is, we show that there are at most $O_ε(m^ε|A|^{2 + 1/3 - 1/1392})$ solutions to the equation $a_1 + a_2 = a_3 + a_4,$ with $a_1, \dots, a_4 \in A$. This improves upon a result of Bourgain and Demeter, and makes progress towards one of their conjectures. A further novelty of our method is that we are able to distinguish between the case of the sphere and the paraboloid in $\mathbb{Z}^4$, since the threshold bound is sharp in the latter case. We also obtain variants of this estimate when $d=3$, where we improve upon previous results of Benatar and Maffucci concerning lattice point correlations. Finally, we use our bounds on additive energies to deliver discrete restriction type estimates for the sphere.

math.NT↗

arXiv2021

Energy estimates in sum-product and convexity problems

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 $C$ are disjoint sets satisfying \[ |\{ (b_1, \dots, b_{2s}) \in B^{2s} \ | \ b_1 + \dots + b_{s} = b_{s+1} + \dots + b_{2s}\}| \ll_{s} |B|^{2s - (\log \log s)^{1/2 - o(1)}} \] and \[ |\{ (c_1, \dots, c_{2s}) \in C^{2s} \ | \ c_1 \dots c_{s} = c_{s+1} \dots c_{2s} \}| \ll_{s} |C|^{2s - (\log \log s)^{1/2 - o(1)}}.\] This generalises previous results of Bourgain--Chang on many-fold sumsets and product sets to the setting of many-fold energies, albeit with a weaker power saving, consequently confirming a speculation of Balog--Wooley. We further use our method to obtain new estimates for $s$-fold additive energies of $k$-convex sets, and these come arbitrarily close to the known lower bounds as $s$ becomes sufficiently large.

math.CO↗

arXiv2020

Difference sets in higher dimensions

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- δ}),\] where $δ>0$ is an absolute constant only depending on $d$. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.

math.CO↗

arXiv2020

Sum-product estimates for diagonal matrices

Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d \times d$ diagonal matrices with real entries. Then for all $δ_1 < 1/3 + 5/5277$, we have \[ |A+A| + |A\cdot A| \gg_{d} |A|^{1 + δ_{1}/d}. \] In this setting, the above estimate quantitatively strengthens a result of Chang.

math.CO↗

arXiv2020

Arithmetic Combinatorics on Vinogradov systems

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 phenomenon.

math.CO↗

arXiv2019

Sums of linear transformations in higher dimensions

In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1 < qs$. Then there exists a constant $δ> 0$ depending only on $q,s$ and $d$ such that for any finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane, we have $$ |q\cdot A + s\cdot A| \geq (|q| +|s|+ 2d-2)|A| - O_{q,s,d}(|A|^{1-δ}) . $$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathscr{L} \in \textrm{GL}_{2}( \mathbb{R})$ be a linear transformation such that $\mathscr{L}$ does not have any invariant one-dimensional subspace of $\mathbb{R}^2$. Then for all finite subsets $A$ of $\mathbb{R}^2$, we have $$ |A + \mathscr{L}(A)| \geq 4|A| - O(|A|^{1-δ}), $$ for some absolute constant $δ> 0$. The main term in this result is sharp as well.

math.CO↗