Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Here $J_3$ is the $3\times3$ all-ones matrix. They did not claim uniqueness. We fully prove their conjecture and classify equality: the maximizers are exactly the simultaneous-permutation conjugates of $A_\star$.
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from th...
Let $N=2^k-1$ and let $\operatorname{wt}(n)$ denote the binary Hamming weight. The Tu-Deng conjecture asserts that, for every $1\le t\le N-1$, at most $2^{k-1}$ pairs $(a,b)\in\{0,\ldots,N-1\}^2$ satisfy $a+b\equiv t\pmod N$ and $\operatorname{wt}(a)+\operatorname{wt}(b)<k$. Partial results are known. We give a complet...
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Fulek defined the $0$-$1$ matrix \[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \] and asked whether $\text{ex}(n,L_3) = O(n)$. We prove that every $r\times s$ $0$-$1$ matrix avoiding $L_3$ has at most $27r+2s$ $1$ entries. Fulek's general lower bound construction has $6n-8$ $1$ entries, so \[ 6...
The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Tur\'an density of $H$ is defined by $\...
Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\lo...
