## 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

### Citations:

Zbl 0712.11037; Zbl 0466.10030
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