A Khintchine-type theorem and solutions to linear equations in Piatetski-Shapiro sequences

Our main result concerns a perturbation of a classic theorem of Khintchine in Diophantine approximation. We give sufficient conditions on a sequence of positive real numbers $({\psi }_n{)}_{n\in \mathbb{N}}$ and differentiable functions $({\phi }_n:J\to \mathbb{R}{)}_{n\in \mathbb{N}}$ so that for Lebesgue-a.e. $\theta \in J$, the inequality $| n\theta + \varphi_n(\theta) | \leq \psi_n$ has infinitely many solutions. The main novelty is that the magnitude of the perturbation $\mathrm{\mid }{\phi }_n(\theta )\mathrm{\mid }$ is allowed to exceed ${\psi }_n$, changing the usual “shrinking targets” problem into a “shifting targets” problem. As an application of the main result, we prove that if the linear equation $y=ax+b$, $a,b\in \mathbb{R}$, has infinitely many solutions in $\mathbb{N}$, then for Lebesgue-a.e. $\alpha >1$, it has infinitely many or finitely many solutions of the form $\lfloor n^{\alpha }\rfloor$ according as $\alpha <2$ or $\alpha >2$.