Equality Cases in the Tu-Deng Conjecture Bound

Abstract

The Tu-Deng Conjecture, made in 2011, asserts that for $k\ge 2$ and $1\le t<2^k-1$, the set \[ S_{t,k}=\left\{(a,b)\in\{0,1,\ldots,2^k-2\}^2: a+b\equiv t\pmod{2^k-1},\quad \wt(a)+\wt(b)<k\right\} \] satisfies \[ |S_{t,k}|\leq 2^{k-1}. \] Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let $r=\wt(t)$, and let $z=k-r$ be the number of zeros in the $k$-bit expansion of $t$. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least $z$. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial $H_W(u,v)$ in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.