Dimension spectrum for a nonconventional ergodic average

TL;DR

This paper calculates the Hausdorff dimension spectrum of binary sequences with prescribed pattern frequencies in nonconventional ergodic averages, revealing the fractal structure of these sets.

Contribution

It introduces a method to compute the dimension spectrum for specific nonconventional averages involving pattern frequencies in binary sequences.

Findings

Determined the Hausdorff dimension of sets with fixed pattern frequency.

Established the dimension spectrum as a function of pattern frequency.

Provided insights into the fractal geometry of nonconventional ergodic averages.

Abstract

We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of $0,1$ sequences, for which the frequency of the pattern 11 in positions $k, 2k$ equals a given number $\theta\in [0,1]$.