A mathematical preprint has identified the asymptotic scale of the smallest positive bias in a binary digit-sum problem when the binary weight is fixed. If F(k) is the infimum of c_t−1/2 over shifts t whose binary representation contains exactly k ones, the result is F(k)∼(1/(2√π))(log_2 k/k)^(3/2) as k grows. In plain terms, the closest possible gap above one-half shrinks with k, on a polynomial-logarithmic scale with a specified leading constant.
The problem concerns integer shifts t, the binary sum-of-digits function s_2(t), and a density called c_t. Fixing s_2(t)=k fixes the number of 1s in the binary representation but not their arrangement, so the result is an optimization over binary patterns with the same weight.
This fixed-weight question is distinct from an earlier pointwise exponential bound quoted in the preprint. That bound concerns one shift at a time, whereas F(k) searches among all shifts with the same binary weight.
The narrow geometry behind the result
The proof brings together three parts: a first-exit argument for configurations with few one-blocks, a high-order expansion for configurations with many blocks, and a rigidity mechanism for near-minimizing binary patterns. Its main analytic tool is a five-cumulant Edgeworth expansion, a high-order approximation that uses cumulants to describe the shape of the relevant distribution.
In the critical block geometry, four normalized cumulants approach 2, 6, 26 and 150. The calculation is expressed through M(t), the number of one-blocks, and D(t), the critical defect. When the defect stays bounded and M(t) grows without bound, the bias is asymptotically D(t)/(8√π) times the reciprocal of M(t) raised to three-halves, up to a uniform smaller-order remainder.
The critical defect has a built-in lower limit: D(t) is at least 4. The proposition governing near-extremal sequences further requires M(t) to grow without bound, the defect to remain bounded, and the number of one-blocks to be no more than about k/log_2 k, in the sense that M(t)≤(1+o(1))k/log_2 k.
Patterns that reach the predicted scale
The lower-bound analysis is matched by an explicit construction for every sufficiently large k. It uses binary patterns with nearly equal, long one-blocks separated by isolated zeros, while keeping s_2(t_k)=k. The resulting bias reaches the leading constant 1/(2√π) on the fixed-weight scale.
The stability result narrows the shape of asymptotic extremizers under its stated condition. Their defect tends to 4, their number of one-blocks is asymptotic to k/log_2 k, and their inner zero-blocks eventually become isolated. For each fixed threshold h≥2, at most two one-blocks have length at most h; the average one-block length is therefore asymptotic to log_2 k.
An infinite isolated-zero model gives the same leading factor in a related asymptotic calculation: its bias is asymptotic to (1/(2√π)) times the reciprocal of N raised to three-halves as N grows. The model therefore agrees with the finite construction at the level of the leading constant.
A bound beyond the fixed-weight problem
The preprint also states a global lower bound for every t≥1: c_t≥1/2+C(log(2+s_2(t))/s_2(t))^(3/2), where C is an absolute positive constant. This extends the result from the fixed-weight extremal question to a lower bound expressed directly through the binary weight.
The central statement is asymptotic, describing the large-k regime rather than giving an explicit finite-k error range. The supplied metadata identifies the work as arXiv version 1, posted on 26 August 2026. No funding source or conflict-of-interest statement is reported in the supplied document or metadata.
Paper data and sources
Original title: Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight
Authors: Kaimin Cheng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text