A Piatetski-Shapiro sequence with exponent $\alpha$ is a of integer parts $n^\alpha$ $(n = 1,2,\ldots)$ non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set those terms. In this article, we study so that equation $ax + by cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give lower bound for its Hausdorff dimension. As corollary, find...

In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solu...