ON SMALL BASES FOR WHICH 1 HAS COUNTABLY MANY EXPANSIONS. Issue 2 (22nd January 2016) - Zou, Yuru; Wang, Lijin; Lu, Jian; Baker, Simon
Abstract : Let q ∈ ( 1, 2 ) . A q ‐expansion of a number x in [ 0, 1 / ( q ‐ 1 ) ] is a sequence ( δ i ) i = 1 ∞ ∈ { 0, 1 } N satisfying x = ∑ i = 1 ∞ δ i q i . Let B ℵ 0 denote the set of q for which there exists x with a countable number of q ‐expansions, and let B 1, ℵ 0 denote the set of q for which 1 has a countable number of q ‐expansions. In Erdős et al [On the uniqueness of the expansions 1 = ∑ i = 1 ∞ q ‐ n i . Acta Math. Hungar. 58 (1991), 333–342] it was shown that min B ℵ 0 = min B 1, ℵ 0 = ( 1 + 5 ) / 2, and in S. Baker [On small bases which admit countably many expansions. J. Number Theory 147 (2015), 515–532] it was shown that B ℵ 0 ∩ ( ( 1 + 5 ) / 2, q 1 ] = { q 1 }, where q 1 ( ≈ 1 . 64541 ) is the positive root of x 6 ‐ x 4 ‐ x 3 ‐ 2 x 2 ‐ x ‐ 1 = 0 . In this paper we show that the second smallest point of B 1, ℵ 0 is q 3 ( ≈ 1 . 68042 ), the positive root of x 5 ‐ x 4 ‐ x 3 ‐ x + 1 = 0 . En route to proving this result, we show that B ℵ 0 ∩ ( q 1, q 3 ] = { q 2, q 3 }, where q 2 ( ≈ 1 . 65462 ) is the positive root of x 6 ‐ 2 x 4 ‐ x 3 ‐ 1 = 0 .