A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets

TL;DR

This paper establishes a strong law of large numbers for complex dependent processes and applies it to analyze the fractal dimensions of sets defined by digit frequency patterns, advancing multifractal analysis.

Contribution

It introduces new conditions for a strong law of large numbers applicable to dependent, mixing processes and uses these to study multifractal properties of digit frequency sets.

Findings

Proved a strong law of large numbers for mixing vector processes.

Derived results on Hausdorff dimensions of digit frequency sets.

Connected probabilistic laws with multifractal formalism.

Abstract

We provide conditions which yield a strong law of large numbers for expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),..., X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polinomial growth and certain regularity properties and $q_i,i>m$ are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places $q_1(n),...,q_\ell(n)$.