Cusick’s conjecture

Cusick’s conjecture asks whether $c_t>\frac12$ for every positive integer $t$, where $c_t$ measures how often adding $t$ does not decrease the binary digit sum. The original conjecture is now the subject of a claimed proof, while the listed problem concerns a sharper quantitative asymptotic.

Known results

• Drmota et al. (2015): $c_t>\frac12$ for a set of $t$ of asymptotic density one.
• Drmota et al. (2015): positivity for integers whose binary expansions contain sufficiently many blocks of ones.
• Drmota et al. (2015): for $t_j=(4^j-1)/3$, $c_{t_j}=\frac12+\frac{\sqrt{3}}{4\sqrt{2\pi j}}+O(j^{-3/2})$, showing the bound is asymptotically tight.

June–August 2026 claimed resolution and refinement

On June 22, 2026, Kaimin Cheng’s preprint A first-exit proof of Cusick’s sum-of-digits conjecture claimed $c_t\ge\frac12+2^{-2s_2(t)-1}$ for every $t\ge1$, hence a complete proof; this remains unverified. A report dated August 26, 2026 described a further claimed fixed-weight asymptotic, $\frac{1}{2\sqrt{\pi}}\left(\frac{\log_2 k}{k}\right)^{3/2}$, but supplied no author or independent confirmation.

Current status (as of August 2026): The original inequality $c_t>\frac12$ is claimed proved by Cheng but remains unverified, and the sharper fixed-weight asymptotic is likewise only claimed; no independent verification or disproof is recorded.