Article
Limit Points of Fractional Parts of Geometric Sequences
Authors
Abstract
Let α > 1 be an algebraic number and ξ a nonzero real num- ber. In this paper, we compute the range of the fractional parts {ξαn} (n = 0, 1, . . .). In particular, we estimate the maximal and minimal limit points. Our results show, for example, that if θ(= 24.97 . . .) is the unique zero of the poly- nomial 2X2 − 50X + 1 with X > 1, then there exists a nonzero ξ∗ satisfying lim supn→∞{ξ∗θn} ≤ 0.02127 . . .. On the other hand, we also prove for any nonzero ξ that lim supn→∞{ξθn} ≥ 0.02003 . . ..
Keywords
Fractional part, limit point, uniform distribution, algebraic number.
Citation
Kaneko, H. (2009). Limit points of fractional parts of geometric sequences. Uniform Distribution Theory, 4(2), 1–37.
H. Kaneko, “Limit points of fractional parts of geometric sequences,” Uniform Distribution Theory, vol. 4, no. 2, pp. 1–37, 2009.
Kaneko H. Limit points of fractional parts of geometric sequences. Uniform Distribution Theory. 2009;4(2):1–37.
Kaneko, H. (2009), ‘Limit points of fractional parts of geometric sequences’, Uniform Distribution Theory, 4(2), pp. 1–37.
Kaneko, Hajime. “Limit Points of Fractional Parts of Geometric Sequences.” Uniform Distribution Theory, vol. 4, no. 2, 2009, pp. 1–37.
Kaneko, Hajime. “Limit Points of Fractional Parts of Geometric Sequences.” Uniform Distribution Theory 4, no. 2 (2009): 1–37.
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Published by: Engineering Journals
