Cusick’s conjecture on the binary sum of digits s(n) of a nonnegative integer n states the following: for all nonnegative integers t we have ct = lim N!1 1 N |{n < N : s(n + t) s(n)}| > 1/2. We prove that for given " > 0 we have ct + ct0 > 1 " if the binary expansion of t contains enough blocks of consecutive 1s (depending on "), where t 0 = 3 · 2 t and is chosen such that 2 ? t < 2+1.
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