Henna Koivusalo

Cut and project sets, linear repetition of patterns, and the Littlewood conjecture

Cut and project sets are, in many senses of the word, regular, but aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. A flexible formalism describes how to translate information on Diophantine approximation to regularity properties of cut and project sets. In this talk I explain recent development of the theory: how to quantify the relationship between Diophantine approximation and regularity properties of cut and project sets. In particular, I give an explicit characterization of linearly repetitive cut and project sets; show that existence of certain types of cut and project sets with a very high regularity is, in fact, equivalent to the Littlewood conjecture from Diophantine approximation being false; and explain some ideas for future research.