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

Abstract

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 (ψn) n∈N and differentiable functions (ϕn : J → R) n∈N so that for Lebesgue-a.e. θ ∈ J, the inequality nθ + ϕn(θ) ≤ ψn has infinitely many solutions. The main novelty is that the magnitude of the perturbation |ϕn(θ)| is allowed to exceed ψn, 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 ∈ R, has infinitely many solutions in N, then for Lebesgue-a.e. α > 1, it has infinitely many or finitely many solutions of the form ⌊n α ⌋ according as α < 2 or α > 2.

On the number of representations of integers as differences between Piatetski-Shapiro numbers

“…Matsusaka and Saito [10] showed that, for all t > s > 2 and a, b, c ∈ N, the set of all α ∈ [s, t] such that the equation ax + by = cz is solvable, has positive Hausdorff dimension. Glasscock [11] showed that if the equation y = ax + b with real numbers a = 0, 1 and b is solvable in N, then, for Lebesgue-a.e. α ∈ (1, 2) (resp.…”

Linear equations with two variables in Piatetski-Shapiro sequences

“…In this article, we discuss the solvability in PS(α) of the equation (1.1) y = ax + b for fixed a, b ∈ R with a / ∈ {0, 1}. Glasscock proved that if equation (1.1) is solvable in N, then for Lebesgue almost every α > 1, it is solvable or not in PS(α) according as α < 2 or α > 2 [Gla17,Gla20]. In addition, as a corollary, he showed that for Lebesgue almost every 1 < α < 2, there are infinitely many (k, ℓ, m) ∈ N 3 such that…”

Linear Diophantine equations in Piatetski-Shapiro sequences

“…, x n ) ∈ PS(α) n satisfying this equation. In this article, we investigate the solvability in PS(α) of linear Diophantine equations (1.1) ax + by = cz for all fixed a, b, c ∈ N. For example, the solvability of the equation y = θx + η for θ, η ∈ R with θ ∈ {0, 1} has been studied by Glasscock [Gla17,Gla20]. He asserts that if the equation y = θx+η has infinitely many solutions (x, y) ∈ N 2 , then for Lebesgue-a.e.…”