Asymptotic repetitive threshold of balanced sequences

Abstract

The critical exponent E ( u ) E(\mathbf u) of an infinite sequence u \mathbf u over a finite alphabet expresses the maximal repetition of a factor in u \mathbf u . By the famous Dejean’s theorem, E ( u ) ≥ 1 + 1 d − 1 E(\mathbf u) \geq 1+\frac 1{d-1} for every d d -ary sequence u \mathbf u . We define the asymptotic critical exponent E ∗ ( u ) E^*(\mathbf u) as the upper limit of the maximal repetition of factors of length n n . We show that for any d > 1 d>1 there exists a d d -ary sequence u \mathbf u having E ∗ ( u ) E^*(\mathbf u) arbitrarily close to 1 1 . Then we focus on the class of d d -ary balanced sequences. In this class, the values E ∗ ( u ) E^*(\mathbf u) are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a d d -ary balanced sequence with the least asymptotic critical exponent for 2 ≤ d ≤ 10 2\leq d\leq 10 .