Digit frequencies of beta-expansions

摘要

Let > 1 be a non-integer. First we show that Lebesgue almost every number has a -expansion of a given frequency if and only if Lebesgue almost every number has infinitely many -expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced -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 (1, 2), there exists a constant c = c() > 0 such that for any p [ 1 2 - c, 1 2 + c], Lebesgue almost every x has infinitely many -expansions with frequency of zeros equal to p.

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