论文标题
piatetski-shapiro序列中有两个变量的线性方程
Linear equations with two variables in Piatetski-Shapiro sequences
论文作者
论文摘要
对于每个非综合$α> 1 $，$ n^α$ $（n = 1,2，\ ldots）$的整数部分的顺序称为piatetski-shapiro序列，带有指数$α$，让$ \ mathrm {ps}（ps}}（α）$表示所有这些条款的集合。对于所有$ x \ subseteq \ mathbb {n} $，我们说方程$ y = ax+b $可在$ x $中解决，如果方程中有许多不同的解决方案$（x，y）\ in x^2 $。让$ a，b \ in \ mathbb {r} $，带有$ a \ neq 1 $和$ 0 \ leq b <a $，并假设方程$ y = ax+b $可在$ \ mathbb {n} $中解决。我们表明，对于所有$ 1 <α<2 $，方程$ y = ax+b $可在$ \ mathrm {ps}（α）$中解决。此外，我们调查了（s，t）$的$α\的集合，以便方程$ y = ax+b $可在$ \ mathrm {ps}（α）$中求解，其中$ 2 <s <t $。最后，我们表明该套装的Hausdorff尺寸与$ 2/s $相吻合。
For every non-integral $α>1$, the sequence of the integer parts of $n^α$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $α$, and let $\mathrm{PS}(α)$ denote the set of all those terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is solvable in $X$ if the equation has infinitely many solutions of distinct pairs $(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\neq 1$ and $0\leq b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<α<2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$. Further, we investigate the set of $α\in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$ where $2< s <t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.
