Asymptotic formula for balanced words

TL;DR

This paper derives asymptotic formulas for counting balanced words with given slope and intercept, linking their distribution to Farey fractions and the Riemann Hypothesis, and employs large sieve inequalities for error estimation.

Contribution

It provides new asymptotic formulas for balanced words, connecting combinatorics with number theory and the Riemann Hypothesis, using advanced sieve techniques.

Findings

Asymptotic formulas for balanced words are established.

Connections between word distribution and Farey fractions are demonstrated.

Error terms are estimated using large sieve inequalities.

Abstract

We give asymptotic formulas for the number of balanced words whose slope $\alpha$ and intercept $\rho$ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case, the error term is deduced using an inequality of large sieve type.