On arithmetic sums of Ahlfors-regular sets

Abstract

Let A,B⊂RA,B⊂R be closed Ahlfors-regular sets with dimensions dimHA=:αdimH⁡A=:α and dimHB=:βdimH⁡B=:β. I prove that dimH[A+θB]≥α+β⋅1−α2−αdimH⁡[A+θB]≥α+β⋅1−α2−α for all θ∈R∖Eθ∈R∖E, where dimHE=0dimH⁡E=0.