Measure and dimension of sums and products

Abstract

We investigate the Fourier dimension, Hausdorff dimension, and Lebesgue measure of sets of the form R Y + Z , RY + Z, where R R is a set of scalars and Y , Z Y, Z are subsets of Euclidean space. Regarding the Fourier dimension, we prove that for each α ∈ [ 0 , 1 ] \alpha \in [0,1] and for each non-empty compact set of scalars R ⊆ ( 0 , ∞ ) R \subseteq (0,\infty ) , there exists a compact set Y ⊆ [ 1 , 2 ] Y \subseteq [1,2] such that dim F ⁡ ( Y ) = dim H ⁡ ( Y ) = dim M ¯ ( Y ) = α \dim _F(Y) = \dim _H(Y) = \overline {\dim _M}(Y) = \alpha and dim F ⁡ ( R Y ) ≥ min { 1 , dim F ⁡ ( R ) + dim F ⁡ ( Y ) } \dim _F(RY) \geq \min \{ 1, \dim _F(R) + \dim _F(Y)\} . This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones. Further, we investigate lower bounds on the measure and dimension of R Y + Z RY+Z for R ⊂ ( 0 , ∞ ) R\subset (0,\infty ) and arbitrary Y Y and Z Z ; these latter results provide a generalized variant of some theorems of Wolff and Oberlin in which Y Y is the unit sphere.