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Effectivization of a theorem of Koksma. (Russian) Zbl 0701.11022
A classical theorem due to J. P. Koksma (1935) states that, if f(t,n) $$(n=1,2,3,...)$$ are real valued, continuously differentiable functions of t defined on the interval [a,b] such that for any natural numbers m, n (m$$\neq n)$$ the function $$f'(t,m)-f'(t,n)$$ is monotonous and satisfies $$| f'(t,m)-f'(t,n)| \geq K>0,$$ where K is a constant independent of t, m and n, then the sequence $$\{f(t,n)\}_{n\geq 1}$$ is uniformly distributed (mod 1) for almost all $$t\in [a,b]$$. Here, it is known also that the discrepancy of the sequence $$\{f(t,n)\}_ n$$ for the first P terms is of $${\mathcal O}(P^{-1/2}(\log P)^{5/2+\epsilon})$$ for every $$\epsilon >0$$ and almost all t [cf. J. F. Koksma and P. Erdős, Indagationes Math. 11, 299-302 (1949; Zbl 0035.321) and J. W. S. Cassels, Proc. Camb. Philos. Soc. 46, 219-225 (1950; Zbl 0035.319)].
The author obtains the following result. Let f(t,n) $$(n=1,2,3,...)$$ be a sequence of functions satisfying the same conditions as specified above. Put $$\alpha =a+\sum^{\infty}_{r=1}a_ r(q_ rh_ r)^{-1},$$ where $$a_ r\geq 0$$, $$q_ r>0$$ and $$h_ r>0$$ are integers defined inductively on r by some complicated formulae, written basically in terms of values of $$f'(a,n)$$, $$f'(b,n)$$ and f(t,n). Then the discrepancy of the particular sequence $$\{f(\alpha,n)\}_ n$$ for the first P terms is of $${\mathcal O}(P^{-1/2}(\log P)^ 4).$$ Thus, one may find for instance a sequence $$\{\alpha^ n\}_ n$$ which is uniformly distributed (mod 1), with an effectively determined constant $$\alpha >1$$.
Reviewer: S.Uchiyama
##### MSC:
 11J71 Distribution modulo one 11K38 Irregularities of distribution, discrepancy 11K06 General theory of distribution modulo $$1$$