We will survey new and old results on the "energy'' of a set of affine transformations and see applications in the geometric side of additive combinatorics, in combinatorial geometry and in questions on growth in the affine group. In particular we will answer to the affirmative a question of Yufei Zhao.
Structure theorems for "approximate groups" show what structure remains when we relax the closure condition for groups. Examples of approximate groups are arithmetic progressions and large subsets of finite groups; a structure theorem shows how an arbitrary approximate group is composed of such examples. We show that if a subset A of GLn(q) is "approximately closed" under multiplication and the group G it generates is soluble, then there are subgroups U and S of G such that: A quickly generates U, S contains most of A, S/U is nilpotent. Briefly: approximate soluble linear groups are (almost) finite by nilpotent. This confirms a conjecture of Helfgott.
We study this famous old problem from the modern perspective of additive combinatorics, and then look at generalizations.
Suppose for each prime p we are given a set Ap (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set Aq of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in Aq will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors.
A complete mapping of a group G is a bijection f : G → G such that the map x f(x) is also a bijection. Hall and Paige conjectured in 1955 that every finite group satisfying a certain necessary condition has a complete mapping; this was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups. I will discuss recent joint work with Freddie Manners and Rudi Mrazovic in which we asymptotically count complete mappings using something like the circle method.
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