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Determinants, Permutations and Additive Combinatorics

Zhi-Wei Sun

Nanjing University



Аннотация: In this talk we will introduce some new problems and related progress on determinants involving Legendre symbols, circular permutations and additive combinatorics. For example, we conjecture that for any finite subset $A$ of an additive cyclic group $G$ with $|A|=n>3$ there is a circular permutation $a_1,\ldots,a_n$ of the elements of $A$ such that all the $n$ sums
$$a_1+a_2+a_3, a_2+a_3+a_4,\ldots, a_{n-2}+a_{n-1}+a_n, a_{n-1}+a_n+a_1, a_n+a_1+a_2$$
are pairwise distinct. The speaker has proved this when $G$ is the infinite cyclic group $\mathbb Z$.


© МИАН, 2024