ABSTRACT

If we add a typical thin set in base a to a typical thin set in base b can we say anything useful about the thickness of the sum set? Some care is needed over the formulation. If, for example, we add the Cantor middle third set to itself then we obviously fill out a complete interval and the problem is of no great interest. On the other hand, a better interpretation of ‘typical base 3 thin set’ is ‘arbitrary set of positive μc measure’ where μc is taken as a uniform probability smeared over the middle third set. It is no longer always the case that the sum contains an interval but one can show, in general, that the sum does have positive Lebesgue measure. More interestingly there is a precise metric result (Brown and Moran, [3]): m   ( E   +   F )   ≥   μ c   ( E ) α   μ c   ( F ) α ,   α   =   log   3 / log   4 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq194.tif"/>