A sequence of positive integers of the form $[n^a]$ for some non-integral a>1 is called a Piatetski-Shapiro sequence (for short PS-sequence). Let $PS(a)$ be the set of all integers of the form [n^a]. In this talk, we discuss the linear equation (E) $x+y=z$ on $PS(a)$. As a main result, we show that for almost all $a >3$, the equation (E) has at most finitely many solutions $(x,y,z)$ which belong to $PS(a)^3$. If time permitted, we give a result for general linear equations and the Hausdorff dimension.
※ 本セミナーは対面で行われます。