Ataleshvara Bhargava

Sharp Transitions for Localized Solutions to a Diophantine Inequality

For fixed $\tau>0$, non-integer $\theta>2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^{\theta}+\cdots +x_s^{\theta} - R|<\tau$ 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...