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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
11J71 Distribution modulo one
11K38 Irregularities of distribution, discrepancy
11K06 General theory of distribution modulo \(1\)