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Digit frequencies of beta-expansions

Abstract : Let β>1 be a non-integer. First we show that Lebesgue almost every number has a β-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many β-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced β-expansions, where an infinite sequence on the finite alphabet {0,1,…,m} is called balanced if the frequency of the digit k is equal to the frequency of the digit m−k for all k∈{0,1,…,m}. Finally we consider variable frequency and prove that for every pseudo-golden ratio β∈(1,2), there exists a constant c=c(β)>0 such that for any p∈[12−c,12+c], Lebesgue almost every x has infinitely many β-expansions with frequency of zeros equal to p.
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https://hal.sorbonne-universite.fr/hal-03113177
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Submitted on : Monday, January 18, 2021 - 10:52:50 AM
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Y.-Q. Li. Digit frequencies of beta-expansions. Acta Mathematica Hungarica, Springer Verlag, 2020, 162 (2), pp.403-418. ⟨10.1007/s10474-020-01032-7⟩. ⟨hal-03113177⟩

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