丢番图不等式局域解的 sharp 相变
Sharp Transitions for Localized Solutions to a Diophantine Inequality
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中文总结 AI 辅助
针对近对角范围变量的丢番图不等式解的存在性，证明了解的个数随参数c呈现 sharp 相变，c>ω时大R必存在解，c<ω时大R可无解，结果类似华林问题的对应结论且可能更强。
中文摘要 AI 辅助
对于固定的 τ>0、非整数 θ>2 和足够大的 s，我们研究丢番图不等式 |x₁^θ+⋯+x_s^θ - R| < τ 在 R→∞ 时的解的个数。此处将变量 x_i 限制在“近对角”范围 X-Y < x_i ≤ X+Y（i=1,…,s），其中 X=(R/s)^(1/θ)，Y≍√X。令 ω=(⌊s/2⌋(θ-1))^(-1/2)，我们将证明：若 Y=c√X 中 c>ω，则当 R 足够大时必然存在解；若 c<ω，则存在任意大的正 R 使得无解。该结果本质上是 sharp 的，仅在 c=ω 处除外。本工作与 Daemen 和 Wright 的结果类似，后者针对华林问题证明了类似结论，不过我们的结果可能比该情形下能得到的结论更强。我们还讨论了其他相关结果及未来可开展的工作。
英文摘要
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.
