The minimal Cusick bias among numbers with exactly k ones in binary is asymptotically (1/(2*sqrt(pi))) (log_2 k / k)^(3/2).
Proof of the TuDeng Conjecture
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abstract
We give a complete proof of the 2011 Tu--Deng conjecture. We begin from its original modular pair-count formulation, prove an equivalent cyclic Hamming weight-drop formulation, and establish the exact transfer identity that connects this count with a two-variable matrix polynomial. The proof then reduces the conjecture to normalized inequalities for the coefficients of that polynomial. A 2011 conjecture by the author which came to be called the Cusick Conjecture (it is a consequence of the Tu--Deng Conjecture) was proved by K. Cheng in 2026. The proof in the present paper extends the cyclic deletion ideas of Cheng. The new ideas might be applicable to other problems.
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The Tu-Deng bound is attained exactly when every gap of zeros in the cyclic binary word contains at least Z-1 ones, resolving Conjecture 3.20 of Flori, Randriambololona, Cohen and Mesnager.
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Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight
The minimal Cusick bias among numbers with exactly k ones in binary is asymptotically (1/(2*sqrt(pi))) (log_2 k / k)^(3/2).
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Cyclic deletion rigidity and Macaulay shadows in the Tu--Deng problem
The Tu-Deng bound is attained exactly when every gap of zeros in the cyclic binary word contains at least Z-1 ones, resolving Conjecture 3.20 of Flori, Randriambololona, Cohen and Mesnager.