Irrationality exponents of generalized Hone series
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Irrationality exponents of generalized Hone series Daniel Duverney1 · Takeshi Kurosawa2
· Iekata Shiokawa3
Received: 1 January 2020 / Accepted: 20 April 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We compute the exact irrationality exponents of certain series of rational numbers, first studied in a special case by Hone, by transforming them into suitable continued fractions. Keywords Irrationality exponent · Continued fraction · Hone’s type expansion Mathematics Subject Classification 11J82 · 11J70
1 Main theorem For a real number α, the irrationality exponent μ (α) is defined by the infimum of the set of numbers μ for which the inequality α − p < 1 q qμ
(1)
has only finitely many rational solutions p/q, or equivalently the supremum of the set of numbers μ for which the inequality (1) has infinitely many solutions. If α is
Communicated by Adrian Constantin.
B
Takeshi Kurosawa [email protected] Daniel Duverney [email protected] Iekata Shiokawa [email protected]
1
Baggio Engineering School, Lille, France
2
Department of Applied Mathematics, Faculty of Science, Tokyo University of Science, Shinjuku, Japan
3
Department of Mathematics, Keio University, Yokohama, Japan
123
D. Duverney et al.
irrational, then μ (α) ≥ 2. If α is a real algebraic irrationality, then μ (α) = 2 by Roth’s theorem [7]. If μ (α) = ∞, then α is called a Liouville number. For every sequence (u n )n≥1 of nonzero numbers or indeterminates, we define u 0 = 1 and θ uk =
u k+1 u k+2 u k , θ 2 u k = θ (θ u k ) = 2 uk u k+1
(k ≥ 0) .
(2)
Theorem 1 Let (xn )n≥1 be an increasing sequence of integers and (yn )n≥1 be a sequence of nonzero integers such that x 1 > y1 ≥ 1 θ 2 xn − θ 2 yn ∈ Z>0 (n ≥ 0) . xn
(3)
Assume that log |yn+2 | = o (log xn ) ,
(i) (ii)
lim inf n→∞
log xn+1 > 2. log xn
Then the series σ =
∞ yn xn
(4)
n=1
is convergent and ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ log xn+1 1 . μ (σ ) = max lim sup , 2+ log xn+1 ⎪ ⎪ log xn ⎪ ⎩ n→∞ ⎭ lim inf n→∞ − 1⎪ log xn Remark 1 The assumption (ii) implies that 2+
1 < 3. log xn+1 lim inf n→∞ −1 log xn
Moreover, if the limit λ := limn→∞ (log xn+1 / log xn ) exists, then μ (σ ) = max λ , 2 +
123
1 , λ−1
Irrationality exponents of generalized Hone series
and so ⎧ √ ⎪ 3+ 5 ⎪ ⎨ λ if λ ≥ = 2.618..., 2 √ μ (σ ) = 3+ 5 1 ⎪ ⎪ ⎩2 + if 2 < λ < . λ−1 2 Hence, under the hypotheses of Theorem 1, μ (σ ) > 2 and therefore σ is transcendental. Examples of series σ satisfying the assumptions of Theorem 1 have been first given by Hone [3] in the case where yn = 1 for every positive integer n, and later by Varona [8] in the case where yn = (−1)n . Both Hone and Varona computed the expansion in regular continued fraction of σ in these special cases and succeeded in proving its transcendence by using Roth’s theorem. For more expansions in regular continued fraction, see also [4–6]. In this paper, we will use basically the same method and transform σ given by (4) into a continued fraction (not regular in general) by using Lemma 2 in Sect. 3. Then w
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