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Quantitative uniform distribution results for geometric progressions. (English) Zbl 1303.11085

The author proves a precise metric result for the discrepancy of sequences of the form \((\{\xi x^{s_n}\})_{n\geq 1}\), where \(\xi\) is an arbitrary fixed positive number and \((s_n)_{n\geq 1}\) an increasing sequence of positive integers. This theorem improves the known metric upper bounds and gives the first metric lower bounds for the discrepancy of such sequences. In particular, this result applies to the geometric progressions \((\{\xi x^{n}\})_{n\geq 1}\). The paper also contains a central limit theorem for sequences of the kind \((f(\xi x^{s_n}))_{n\geq 1}\) where \(f\) is a function such that \(f(x)=f(x+1)\), \(\int_0^1 f(x) dx=0\) and \(\mathrm{Var}_{[0,1]}f\leq 2\). Even if the geometric progressions are somehow similar to lacunary sequences, the techniques used for lacunary function systems cannot be easily adapted to the case of geometric progressions. Indeed, the proof of the main result for sequences of the kind \((\{\xi x^{s_n}\})_{n\geq 1}\) is based on new ideas exploiting their more inhomogeneous structure. For instance, instead of the orthogonality properties typical of lacunary sequences, the author uses the highly oscillatory behaviour of \(f(\xi x^n)\) w.r.t. \(f(\xi x^m)\) for \(n\gg m\) and the van der Corput inequality in calculating integrals. However, the speed of oscillation of \(f(\xi x^n)\) when \(x\) increases makes martingale approximations much more complicated than in the lacunary case.

MSC:

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