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Discrete Brunn-Minkowski Inequality for subsets of the cube
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abstract
We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq |A|^{1/p}|B|^{1/q}|V|^{\log_{2}(p^{1/p}q^{1/q})} $$ holds for all $p \in (1, \infty)$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems.
Forward citations
Cited by 2 Pith papers
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Optimal Young's convolutions inequality and its reverse form on the hypercube
Sharp Young's convolution inequality and its reverse on {0,1}^d are proved, with optimal diagonal exponent p_r = 2r/log_2(2+2^r).
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Inequalities in Fourier analysis on binary cubes
For functions supported on the binary cube, the full range of exponents in the sharp dimension-free Hausdorff-Young and diagonal Young inequalities is characterized exactly.
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