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Explicit lower bounds for \(\| (3/2)^k\|\). (English) Zbl 1126.11325
Summary: Let \(\| (3/2)^k\|\) denote the distance from \((3/2)^k\) to the nearest integer. F. Beukers [Math. Proc. Camb. Philos. Soc. 90, 13–20 (1981; Zbl 0466.10030)] proved that \(\| (3/2)^k\|>2^{-0.9\,k}\) for \(k\geq 5\) and A. K. Dubickas [Russ. Math. Surv. 45, No. 4, 163–164 (1990); translation from Usp. Mat. Nauk 45, No. 4 (274), 153–154 (1990; Zbl 0712.11037)] showed the better inequality \(\| (3/2)^k\|>(0.5769)^k\) for \(k\) large enough. In this paper we improve the constant \(0.5769\) to \(0.5770173776\ldots\), by refining Dubickas’ computations. We also prove that \(\| (3/2)^k\|>2^{-0.8\,k}\) for \(k\geq5\).

MSC:
11J04 Homogeneous approximation to one number
11P05 Waring’s problem and variants
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