Given probability distributions $\mu$ and $\nu$ on measure spaces $X$ and $Y$, and a closed set $S \subseteq X \times Y$, when is there a probability distribution on $X \times Y$ whose marginals are $\mu$ and $\nu$, and whose support is precisely $S$? We answer the question when the marginals are discrete, and when the marginals are continuous distributions on the real line. Of special interest is the case where $S \subseteq [0,1]^2$ and $\mu$ and $\nu$ are Lebesgue measure; then the above question is tantamount to ``when is $S$ the support of a doubly stochastic measure?". The discrete case is generalized to determine when a (possibly infinite) edge-capacitated, node-weighted graph supports a full, nowhere-zero flow; for the continuous case we provide a particularly straightforward characterization when the set in question is regular (i.e., is the closure of its interior).
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quant...
Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $\mu$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of w...
Collapsing a $T^{\kappa^+}_{\omega_1}$-Ramsey cardinal $\kappa$ to $\omega_2$ gives, for every countable coloring of $[\omega_2]^2$, a stationary set $X$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. The color-$i$ graph on $X$ has diameter at most two afte...
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance,...
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
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