A lower bound for Cusick’s conjecture on the digits of n + t

Source abstract

Abstract Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t , define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that c t &gt; 1/2. We have the elementary bound 0 &lt; c t &lt; 1; however, no bound of the form 0 &lt; α ≤ c t or c t ≤ β &lt; 1, valid for all t , is known. In this paper, we prove that c t &gt; 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).