On the morphology of 𝛾-expansions with deleted digits

Keane, Mike; Smorodinsky, Meir; Solomyak, Boris

doi:10.1090/S0002-9947-1995-1290723-X

On the morphology of $\gamma$-expansions with deleted digits
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by Mike Keane, Meir Smorodinsky and Boris Solomyak PDF

Trans. Amer. Math. Soc. 347 (1995), 955-966 Request permission

Abstract:

We investigate the size of the set of reals which can be represented in base $\gamma$ using only the digits $0,1,3$. It is shown that this set has Lebesgue measure zero for $\gamma \leqslant 1/3$ and equals an interval for $\gamma \geqslant 2/5$. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of $(1/3,2/5)$.

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