Speaker：Yiyu Liang（Beijing Jiaotong University ）

Time：2022-05-19, 15:00

Location：Tencent Meeting ID： 301497640（No Password）

Abstract:

The results in this talk are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. We find a Salem set of dimension 1 that contains no nontrivial solution of any of these equations. Variants of this construction can also be used to obtain Salem sets that avoid solutions of translation-invariant linear equations of other kinds, for instance, when the collection of linear equations to be avoided is uncountable or has irrational coefficients. While such constructions seem to suggest that Salem sets can avoid many configurations, our second type of results offers a counterpoint. We show that a set whose Fourier dimension exceeds 2/(v+1) cannot avoid nontrivial solutions of all equations of some particular form.