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

Let $s$ be the sum-of-digits function in base $2$, which returns the number of $\mathtt 1$s in the base-2 expansion of a nonnegative integer. For a nonnegative integer $t$, define the asymptotic density $$c_t=\lim_{N\rightarrow \infty} \frac 1N\bigl\lvert\{0\leq n<N:s(n+t)\geq s(n)\}\bigr\rvert.$$ T.~W.~Cusick conjectured that $c_t>1/2$. We have the elementary bound $0<c_t<1$; however, no bound of the form $0<\alpha\leq c_t$ or $c_t\leq \beta<1$, valid for all $t$, is known. In this paper, we prove that $c_t>1/2-\varepsilon$ as soon as $t$ contains sufficiently many blocks of $\mathtt 1$s 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).