Normal numbers in sparse Cantor sets

Abstract

We consider Cantor-type sets of Hausdorff dimension zero, consisting of all numbers whose base-2 expansion can have a 1 only at positions belonging to a given sparse set (local count at least log k in every interval of length k). We prove that the measure induced by independent, non-identically distributed Bernoulli digits assigns full mass to numbers that are normal in every odd base. The proof extends Schmidt's 1960 method to this Hausdorff zero-dimensional setting, and we provide an explicit algorithmic construction of such numbers -- yielding the first known examples of numbers deterministic in base~2 yet normal in all odd bases. This work supports our broader conjecture that given determinism in one base, normality in all multiplicatively independent bases is prevalent.