An official website of the United States government
Gowers, Tim
(Ed.)
We prove that for $$d\geq 0$$ and $$k\geq 2$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$higher energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$$) is at most $$|A|^{\log_{2}(2^k+2)}$$, and $$\log_{2}(2^k+2)$$ is the best possible exponent. We also show that if $$d\geq 0$$ and $$2\leq k\leq 10$$, for any subset $$A$$ of a discrete cube $$\{0,1\}^d$$, the $k-$additive energy of $$A$$ (i.e., the number of $2k-$tuples $$(a_1,a_2,\dots,a_{2k})$$ in $$A^{2k}$$ with $$a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$$) is at most $$|A|^{\log_2{ \binom{2k}{k}}}$$, and $$\log_2{ \binom{2k}{k}}$$ is the best possible exponent. We discuss the analogous problems for the sets $$\{0,1,\dots,n\}^d$$ for $$n\geq2$$.
more »
« less
Full Text Available