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
ct > 1/2. We have the elementary bound 0 <
ct < 1; however, no bound of the form 0 <
α ≤
ct or
ct ≤
β < 1, valid for all
t, is known. In this paper, we prove that
ct > 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).