On the Number of Unique Expansions in Non-integer Bases
نویسنده
چکیده
Let q > 1 be a real number and let m = m(q) be the largest integer smaller than q. It is well known that each number x ∈ Jq := [0, P ∞ i=1 mq ] can be written as x = P ∞ i=1 ciq −i with integer coefficients 0 ≤ ci < q. If q is a non-integer, then almost every x ∈ Jq has continuum many expansions of this form. In this note we consider some properties of the set Uq consisting of numbers x ∈ Jq having a unique representation of this form. More specifically, we compare the size of the sets Uq and Ur for values q and r satisfying 1 < q < r and m(q) = m(r).
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