arXiv subjects

Ataleshvara Bhargava

Publications and source records attributed to Ataleshvara Bhargava.

5 recordsLinked to original sources

arXiv2026

Sharp Transitions for Localized Solutions to a Diophantine Inequality

For fixed $τ> 0$, non-integer $θ> 2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^θ+\cdots +x_s^θ - R| < τ$ as $R \to \infty$. Here, we restrict the variables $x_i$ in the ``almost diagonal" range $X-Y < x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/θ}$ and $Y \asymp \sqrt{X}$. Let $ω= (\lfloor s/2 \rfloor(θ-1))^{-1/2}$. We will show that if $Y = c\sqrt{X}$ for some $c > ω$ then for sufficiently large $R$ there must always exist solutions, but if $c < ω$ then there exist arbitrarily large positive $R$ for which there are no solutions. Our result is thus essentially sharp, with the exception of $c = ω$. This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting. We discuss other related results and further work to be done.

math.NT↗

arXiv2026

Mean Value Estimates for a Real-Exponent Analogue of Waring's Problem

For non-integer $θ> 3$ and $κ\geq 1$, we show that the smallest $r_0$ such that the mean value estimate \[ \int_{-κ}^κ \Big| \sum_{X < x \leq 2X} e(αx^θ) \Big|^{2r} dα\ll_ε κX^{2r - θ+ε} \] holds for all integers $r \geq r_0$ satisfies $2r_0 \leq θ^2(1+O(θ^{-1/2}))$. This is an improvement over the previous bound by Poulias of $2r_0 \leq (\lfloor 2θ\rfloor + 1)(\lfloor 2θ\rfloor + 2)$. As a consequence, the bound on the asymptotic order of the minimum number of variables required to prove the expected asymptotic formula for the number $R_{s,θ}(N)$ of solutions $(x_1,\ldots,x_s) \in \mathbb{N}^s$ to the Diophantine equation \[ \lfloor x_1^θ \rfloor+\cdots+\lfloor x_s^θ \rfloor = N \] is improved by a factor of $4$. We also discuss a certain Diophantine system which arises naturally from our proof, which may have applications to other counting problems and may be of independent interest.

math.NT↗

arXiv2026

An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents

We prove an approximate analog of Vu's Theorem on economical bases of $k$th powers for non-integer powers. Fix a non-integer $θ> 2$ and real $τ> 0$. Let $s \geq θ^2+9θ^{3/2}+2 $ if $θ> 3$ and $s \geq (\lfloor 2θ\rfloor+1)(\lfloor 2θ\rfloor+2)+1$ if $2 < θ< 3$. We show the existence of a set $\mathfrak{X} \subseteq \mathbb{N}$ such that the number $R_{\mathfrak{X},s,θ,τ}(Λ)$ of integer solutions $(x_1,\ldots,x_s) \in \mathfrak{X}^s$ to the equation \begin{align*} |x_1^θ +\cdots+ x_s^θ - Λ| < τ\end{align*} satisfies $R_{\mathfrak{X},s,θ,τ}(Λ) \asymp τ\log(Λ)$ for all sufficiently large real $Λ> 0$.

math.NT↗

arXiv2025

Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces

In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in $\bigcap_{ε\in (0,1)} L^{2-ε}(\mathbb{T}^2) \cap \dot H^{-ε}(\mathbb{T}^2)$. Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemarié-Rieusset, who proved a similar result in the space $\dot{H}^{-1} \cap {BMO}^{-1}$. The main new ingredient is the incorporation of intermittency into the construction of the solutions.

math.AP↗

arXiv2024

A study guide for the $\ell^2$ decoupling theorem for the paraboloid

This article serves as a study guide for the $\ell^2$ decoupling theorem for the paraboloid originally proved by Bourgain and Demeter. Given its popularity and importance, many expositions about the $\ell^2$ decoupling theorem already exist. Our study guide is intended to complement and combine these existing resources in order to provide a more gentle introduction to the subject.

math.CA↗