On the correlation of $ k $ symbols

Abstract

In 2002 Mauduit and Sárközy started to study finite sequences of $ k $ symbols<disp-formula> <label/> <tex-math id="FE1"> $ E_{N} = \left(e_{1}, e_{2}, \cdots, e_{N}\right)\in \mathcal{A}^{N}, $ </tex-math></disp-formula>where<disp-formula> <label/> <tex-math id="FE2"> $ \mathcal{A} = \left\{a_{1}, a_{2}, \cdots, a_{k}\right\}, \ \ (k\in \mathbb{N}, k\geq 2) $ </tex-math></disp-formula>is a finite set of $ k $ symbols. Bérczi estimated the pseudorandom measures for a truly random sequence $ E_{N} $ of $ k $ symbol. In this paper, we shall study the minimal values of correlation measures for the sequences of $ k $ symbols, developing the methods similar to those introduced by Alon, Anantharam, Gyarmati, and Schmidt, among others.