Hausdorff dimensions of Beatty multiple shifts

2025/07/15 by Ban, Jung-Chao, Hu, Wen-Guei, Lai, Guan-Yu
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.10982

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.

Citations

• Hausdorff dimensions of affine multiplicative subshifts
• Dimensions of "self-affine sponges" invariant under the action of multiplicative integers
• Multifractal analysis of weighted ergodic averages
• Some Aspects of Multifractal analysis
• Nonlinear large deviations
• Thermodynamic formalism and large deviations for multiplication-invariant potentials on lattice spin systems
• Multifractal analysis of some multiple ergodic averages
• Pattern generation problems arising in multiplicative integer systems
• Nonconventional averages along arithmetic progressions and lattice spin systems
• Multifractal Analysis of Multiple Ergodic Averages
• Level sets of multiple ergodic averages

Related

• Hausdorff dimensions of affine multiplicative subshifts
• Hausdorff dimension of specification for the (α,β)-shifts
• The Multiplicative golden mean shift has infinite Hausdorff measure
• Hausdorff dimension of frequency sets in beta-expansions
• Hausdorff dimension of the parameters for (α,β)-transformations with the specification property