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Effectivization of a lower bound for $$\| (4/3)^k\|$$. (English. Russian original) Zbl 1230.11088
Math. Notes 85, No. 6, 877-885 (2009); translation from Mat. Zametki 85, No. 6, 927-935 (2009).
Waring’s problem is concerned with the representations of positive integers as sums of $$k$$th powers, i.e., with the solutions of the Diophantine equation $x_1^k + x_2^k + \dots + x_s^k = N, \tag{1}$ where $$N$$ is a fixed positive integer and $$x_1, \dots, x_s$$ are non-negative integer unknowns. This problem has two very different versions. First, one may ask what is the least $$s$$ such that all $$N \geq 1$$ can be represented in the form (1). The least $$s$$ with this property is usually denoted $$g(k)$$. Second, one may ask what is the least $$s$$ such that all sufficiently large integers $$N$$ can be represented in the above form; the least such $$s$$ is usually denoted $$G(k)$$. The estimation of $$G(k)$$ is one of the central problems in additive number theory and has been a driving force behind the development of the circle method for the past ninety years. The value of $$g(k)$$, on the other hand, turns out to be determined by the arithmetic properties of certain relatively small $$N$$, and its study leads to some interesting questions on Diophantine approximation.
The paper under review establishes two inequalities related to the study of $$g(k)$$. The value of $$g(k)$$ depends on the inequality $$\| (3/2)^k \| \geq (3/4)^k$$, where $$\| x \|$$ denotes the distance from $$x$$ to the nearest integer. Thus, several authors have given estimates of the form $\left\| (3/2)^k \right\| \geq C^k \qquad\text{for all integers } k \geq k_0,$ with explicit values of $$C$$ and $$k_0$$. In particular, the best result to date has been obtained by V. Zudilin [J. Théor. Nombres Bordx. 19, No. 1, 311–323 (2007; Zbl 1127.11049)], who gave such a bound with $$C = 0.5803$$. While the value of $$k_0$$ in Zudilin’s work is effectively computable, it is not easy to compute it. In this paper, the author uses Zudilin’s method to obtain a fully explicit, albeit slightly weaker, estimate: $\left\| (3/2)^k \right\| \geq (0.5795)^k \qquad\text{for all integers } k \geq 871,387,440,264.$ He further shows that $\left\| (4/3)^k \right\| \geq (0.491)^k \qquad\text{for all integers } k \geq k_1,$ where $$k_1$$ is an explicitly given number of the order of $$5.868 \times 10^{18}$$. This is also a fully explicit version of a result of Zudilin [op. cit.].

MSC:
 11J54 Small fractional parts of polynomials and generalizations 11J25 Diophantine inequalities
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References:
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