Discrepancy bounds for normal numbers generated by necklaces in arbitrary base

Mordechay B. Levin (1999) has constructed a number λ which is normal in base 2, and such that the sequence ${\left(\left\{2^n\lambda \right\}\right)}_{n=0,1,2,\dots }$ has very small discrepancy $N\cdot D_N=O\left({(\log N)}^2\right)$. This construction technique was generalized by Becher and Carton (2019), who generated normal numbers via nested perfect necklaces, for which the same upper discrepancy estimate holds. In this paper we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for Levin's normal number in arbitrary prime base p this upper bound for the discrepancy is best possible. This result generalizes a previous result by the authors (2022) in base 2.

Our result for Levin's normal number in any prime base might support the guess that $O\left({(\log N)}^2\right)$ is the best order in N that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds.