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Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are exponentially smaller than the digital nets. We also establish a lower bound on the size of such almost nets.
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an alm
We show that if $fcolon S_n to {0,1}$ is $epsilon$-close to linear in $L_2$ and $mathbb{E}[f] leq 1/2$ then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and moreover this is sharp: any such union is close to linear. This constitute
We give a characterization of all matrices $A,B,C in mathbb{F}_{2}^{m times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,Cinmathbb{F}_{2}^{mathbb{N}timesmathbb{N}}$ which generate a $(0,2)$-sequence in base $2$.
Let $mathcal{F}$ and $mathcal{G}$ be two $t$-uniform families of subsets over $[k] = {1,2,...,k}$, where $|mathcal{F}| = |mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $mathcal{F}$ and $mat