Sharp Transitions for Localized Solutions to a Diophantine Inequality

Abstract

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&#34; 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&#39;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.