Linear Diophantine equations in Piatetski-Shapiro sequences

Abstract

Let x denote the integer part of x ∈ R. For a non-integral α > 0, the sequence ( n α ) ∞ n=1 is called the Piatetski-Shapiro sequence with exponent α. Let PS(α) = { n α : n ∈ N}. We say that an equation f (x 1 , . . . , x n ) = 0 is solvable in PS(α) if there are infinitely many pairwise distinct tuples (x 1 , . . . , 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. α > 1 it is solvable or not in PS(α) according as α < 2 or α > 2. As a direct consequence, for Lebesgue-a.e. 1 < α < 2, the equation z = (a/c)x + (b/c) is solvable in PS(α) for all a, b, c ∈ N with gcd(a, c) | b. In other words, the equation (1.1) with gcd(a, c) | b is solvable in PS(α). On the other hand, for α > 2, we did not know at all whether the equation (1.1) is solvable in PS(α) or not.Our main result provides an answer to this question. We consider the set of α in a short interval [s, t] ⊂ (2, ∞) such that (1.1) is solvable. The following theorem asserts that the Hausdorff dimension of this set is positive. To state the theorem, let {x} be the fractional part of x ∈ R, and dim H (X) the Hausdorff dimension of X ⊆ R (the definition will be recalled in Section 2).

A system of certain linear Diophantine equations on analogs of squares

“…Hence, we deduce the positivity of the lower limit of #T ≤x (α)/x. This idea comes from the work of Matsusaka and the second author [MS21]. They showed that for any fixed a, b, c ∈ N, there are uncountably many α > 2 such that the linear equation ax + by = cz has infinitely many solutions (x, y, z) ∈ PS(α) 3 .…”

A system of certain linear Diophantine equations on analogs of squares

“…Hence, we conclude the positiveness of the limit inferior of #T ≤x (α)/x. This idea comes from the work of Matsusaka and the second author [MS21]. They showed that for any fixed a, b, c ∈ N, there are uncountably many α > 2 such that the linear equations ax + by = cz has infinitely many solutions (x, y, z) ∈ PS(α) 3 .…”

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

“…, x k ) = 0. 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.…”