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We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM 20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant appro
We prove upper bounds on deterministic communication complexity in terms of log of the rank and simp
Let $f: {0,1}^n to {0, 1}$ be a boolean function, and let $f_land (x, y) = f(x land y)$ denote the AND-function of $f$, where $x land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_land$ and show that, up to a $log
Hedetniemi conjectured in 1966 that $chi(G times H) = min{chi(G), chi(H)}$ for all graphs $G$ and $H$. Here $Gtimes H$ is the graph with vertex set $ V(G)times V(H)$ defined by putting $(x,y)$ and $(x,y)$ adjacent if and only if $xxin E(G)$ and $yyin
We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there exists an af