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A sum-product estimate in finite fields, and applications
AbstractLet A be a subset of a finite field $$ F := \mathbf{Z}/q\mathbf{Z} $$ for some prime q. If $$ |F|^{\delta} < |A| < |F|^{1-\delta} $$ for some δ > 0, then we prove the estimate $$ |A + A| +Expand
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New Bounds on cap sets
We provide an improvement over Meshulam's bound on cap sets in $F_3^N$. We show that there exist universal $\epsilon>0$ and $C>0$ so that any cap set in $F_3^N$ has size at most $C {3^N \overExpand
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Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type
In this paper we investigate three unsolved conjectures in geomet- ric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. WeExpand
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Finite time blow-up for a dyadic model of the Euler equations
We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In theExpand
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Garaev's inequality in Finite Fields not of prime order
In the present paper, we extend Garaev’s techniques to the set of fields which are not necessarily of prime order. Our goal here is just to find an explicit estimate in the supercritical settingExpand
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A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation
Abstract. We prove that for the Navier—Stokes equation with dissipation $ (-\Delta)^\alpha $ where 1 < α < 5 /4, and smooth initial data, the Hausdorff dimension of the singular set at time of firstExpand
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Maximal operators over arbitrary sets of directions
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New bounds for Kakeya problems
We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.
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Kakeya sets in Cantor directions
We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union ofExpand
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An improved bound on the Minkowski dimension of Besicovitch sets in $\mathbb{R}^3$
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorfi dimensions of such a set must be greater than or equal to 5= 2i n 3 . InExpand
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