论文标题
piatetski-shapiro序列中的线性双方方程
Linear Diophantine equations in Piatetski-Shapiro sequences
论文作者
论文摘要
带有指数$α$的Piatetski-shapiro序列是$ n^α$ $（n = 1,2，\ ldots）$的整数序列，具有非整合$α> 0 $。我们让$ \ mathrm {ps}（α）$表示这些术语的集合。在本文中，我们研究了$α$的集合，以便方程式$ ax + by = cz $具有无限的许多成对差异解决方案$（x，y，z）\ in \ mathrm {ps}（α）^3 $，并为其hausdorff dimension提供了下限。作为推论，我们发现了许多$α> 2 $，因此$ \ mathrm {ps}（α）$包含无限的许多算术进程，长度为$ 3 $。
A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.
