Lacunarity and the Bohr topology

Abstract

If G is an abelian group, then G^# denotes G equipped with the weakest topology that makes every character of G continuous. This is the Bohr topology of G. If G=ℤ, the additive group of the integers and A is a Hadamard set in ℤ, it is shown that: (i) A−A has 0 as its only limit point in ℤ^#; (ii) no Sidon subset of A−A has a limit point in ℤ^#; (iii) A−A is a Λ(p) set for all p<∞. This leads to an explicit example of a set which is Λ(p) for all p<∞ and is dense in ℤ^#. If f(x) is a quadratic or cubic polynomial with integer coefficients, then the closure of f(ℤ) in the Bohr compactification of ℤ is shown to have Haar measure 0. Every infinite abelian group G contains an I₀ set A of the same cardinality as G such that 0 is the only limit point of A−A in G^#.