| Abstract: We discuss two related notions of ``approximate subgroups" inside finite sets of integers: Bohr sets, which capture simultaneous Diophantine approximation, and symmetric generalized arithmetic progressions (GAPs). For example, fix natural numbers $N$ and $d$ and consider the following pair of questions: (1) For fixed $\alpha_1,\dots,\alpha_d\in \mathbb R$, how close to an integer can we
simultaneously make $n^2\alpha_1,\dots,n^2\alpha_d$ for some $1\leq n \leq N$? (2) How large can a set of the form $\{x_1\ell_1+\cdots+x_d\ell_d: |\ell_i|\leq L_i\}\subseteq [-N,N]$ be before it is guaranteed to contain a perfect square? Our discussions range from classical facts like the Kronecker approximation theorem and Linnik's theorem, to a recent breakthrough result of Maynard and its potential future applications. In between we survey results including previous joint work with Neil Lyall and Ernie Croot. |