Substitutions and 1/2-discrepancy of $\{n \theta + x\}$

TL;DR

This paper studies the behavior of 1/2-discrepancy sums of sequences generated by irrational rotations, using substitution methods, and demonstrates their possible growth rates and value ranges, especially for badly approximable numbers.

Contribution

It introduces a substitution-based approach to analyze 1/2-discrepancy sums and establishes new results on their growth and value distribution for specific irrational rotations.

Findings

Discrepancy sums can achieve any non-trivially forbidden asymptotic growth rate.

For badly approximable $ heta$, the value range of the sequence is asymptotically logarithmic.

The approach provides stronger bounds than the classical Denjoy-Koksma inequality.

Abstract

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to $x=0$. The first application is to show that any asymptotic growth rate of the discrepancy sums not trivially forbidden may be achieved. A second application is to show that for badly approximable $\theta$ and any $x$ the range of values taken over $i=0,1,...n-1$ is asymptotically similar to $\log(n)$, a stronger conclusion than given by the Denjoy-Koksma inequality.