Ataleshvara Bhargava

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20212026

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math.NT2026

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

Ataleshvara Bhargava

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 $s \geq θ^{2} + 9 θ^{3/2} + 2$ if…

math.NT2026

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

Ataleshvara Bhargava

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…

math.NT2026

Sharp Transitions for Localized Solutions to a Diophantine Inequality

Ataleshvara Bhargava

For fixed $τ > 0$ , non-integer $θ > 2$ and large enough $s$ , we investigate the number of solutions to the Diophantine inequality $∣ x_{1}^{θ} + \dots + x_{s}^{θ} - R ∣ < τ$ as $R \to \infty$ . H…