AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A \subset \{1, \ldots, n\}$ such that $|A + A| \leq m$, where $k = \Theta(\log n)$ and $m \leq k^{1 + \alpha}$, for small $\alpha>0$, answering a question of Green and Morris.
Let $q_n$ denote the $n$th product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every $\varepsilon>0$ and all sufficiently large integers $\rho$, \[ \liminf_{n\to\infty}(q_{n+\rho}-q_n) \leq \exp\left((\p...
A randomized algorithm returns a $(1\pm\varepsilon)-approximation with failure probability at most $\delta$ in $O^*(2^k\varepsilon^{-2}\log(1/\delta)$ time.
We show that set cover on a universe of size $n$ and with sets of size at most $k$ can be solved in time $2^{(1-1/k+O(1/k^{3/2}))n}$. This improves on a $2^{(1-0.929/k)n}$-time algorithm of Bj\"orklund (STACS 2010) for all sufficiently large $k$.
We describe, with proof, an infinite class of examples of finite sets of integers $A$ for which $\|\widehat{1_A}\|_1\geq (c_\mathcal{R}-o_{|A| \rightarrow \infty}(1))\sqrt{|A|}$ where $c_\mathcal{R}=0.953\dots$.
For $q>p$, let $T(n,q,p)$ be the minimum number of translates of the cube \(\{0,1,\dots,p-1\}^n\) required to cover the $n$-dimensional torus $(\mathbb{Z}/q\mathbb{Z})^n$. We show that for each $q$ there exists a constant $1\le \Lambda_q \le 2$ such that $T(n,q,2)=(\Lambda_q + o(1))(q/2)^n$.
Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{89}{90} + o(1)}$.
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