Deficit bounds and equality cases in the Tu--Deng problem

Abstract

Let $N=2^k-1$, and let $S_{t,k}$ consist of the pairs $0\le a,b<N$ such that $a+b\equiv t\pmod N$ and $\wt(a)+\wt(b)<k$. The Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$ has recently been proved. In this paper, we give a combinatorial proof of the equality criterion. If the $k$-bit cyclic word of $t$ has $Z$ zeros and $g_1,\ldots,g_Z$ are the numbers of ones between successive zeros, then $$|S_{t,k}|=2^{k-1} \quad\Longleftrightarrow\quad g_i\ge Z-1\quad(1\le i\le Z).$$ Moreover, we recover the resulting enumeration of the equality parameters. Beyond equality, if $R\ge Z\ge2$, where $R$ is the number of ones, then every nonequality parameter satisfies $$2^{k-1}-|S_{t,k}|\ge 2^{R-Z+1},$$ and we classify all cases in which this bound is attained. For $R<Z$ we obtain a congruence for $|S_{t,k}|$ and a lower bound for the deficit in terms of the number of cyclic runs of ones.