Hausdorff dimensions of Beatty multiple shifts

TL;DR

This paper introduces the Beatty multiple shift, generalizing existing shifts, and derives formulas for their Hausdorff and Minkowski dimensions, linking them to classical number theory concepts.

Contribution

It presents a new generalization called the Beatty multiple shift and provides explicit formulas for their Hausdorff and Minkowski dimensions.

Findings

Derived formulas for Hausdorff and Minkowski dimensions.

Connected dimension formulas to classical number theory.

Extended previous shift models to a more general framework.

Abstract

In this paper, the Beatty multiple shift is introduced, which is a generalization of the multiplicative shift of finite type (multiple SFT) [Kenyon, Peres and Solomyak, Ergodic Theory and Dynamical Systems, 2012] and the affine multiple shift [Ban, Hu, Lai and Liao, Advances in Mathematics, 2025]. The Hausdorff and Minkowski dimension formulas are obtained, and the coefficients of the formula is closely related to the classical disjoint covering of the positive integers in number theory.