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ON CONFIGURATIONS WHERE THE LOOMIS–WHITNEY INEQUALITY IS NEARLY SHARP AND APPLICATIONS TO THE FURSTENBERG SET PROBLEM

Part of: Designs and configurations Finite fields and commutative rings (number-theoretic aspects) Extremal combinatorics Discrete geometry

Published online by Cambridge University Press:  07 January 2015

Ruixiang Zhang
Ruixiang Zhang*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08540, U.S.A. email ruixiang@math.princeton.edu
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Abstract

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

MSC classification

Primary: 11T99: None of the above, but in this section 52C17: Packing and covering in $n$ dimensions
Secondary: 05B25: Finite geometries 05D99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 145 - 161
Copyright
Copyright © University College London 2015 

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