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.
For $N\geq 2$ and $k\geq 1$, let $M_k(N):=\#\{x_1\cdots x_k : x_i\in\{1,\ldots,N\}\text{ for all } i\}$ be the $k$-dimensional multiplication table. Given $N$, Khovanskii's theorem implies that $M_k(N)$ agrees, for all sufficiently large $k$, with a polynomial in $k$ of degree $\pi(N)$. We determine the asymptotic size...
For $n\ge5$, let $F\colon\mathbb{F}_2^n\to \mathbb{F}_2^n$ be almost perfect nonlinear and write $N=2^n$. It is proven that the linearity $\mathcal{L}(F)$ of $F$, i.e., the largest absolute Walsh coefficient of a nonzero component, is at most $N-10$ in even dimension and at most $N-6$ in odd dimension. This improves th...
Let $\mathbb N$ denote the set of nonnegative integers. For an integer $k>1$ and a set $A\subseteq\mathbb N$, let $R_{1,k}(A,n)$ denote the number of solutions of $n=a_1+ka_2$ with $a_1,a_2\in A$. For integers $k>1$ and $t\ge1$, Yang and Chen defined $f_k(t)$ to be the number of sets $A\subseteq\mathbb N$ for which \[...
Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cyc...
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
Let $A$ be a stochastic $n\times n$ matrix with $\operatorname{rank}A\leq k$, where $1\leq k\leq n$. Write $n=qk+s$, where $0\leq s<k$. The author conjectured in 2018 that $\operatorname{per}A\leq (q!/q^q)^{k-s}((q+1)!/(q+1)^{q+1})^s$, with equality if and only if $A=P(J_q^{\oplus(k-s)}\oplus J_{q+1}^{\oplus s})Q$, whe...
Yair Lavi· 0 citations
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