Ubiquity of simplices in subsets of vector spaces over finite fieldsО симплициальной повсеместности в подмножествах векторных пространств над конечными полями

We prove that a sufficiently large subset of the d-dimensional vector space over a finite field with q elements, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{F} $$\end{document}qd, contains a copy of every k-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.