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A lower bound for the diaphony of generalised van der Corput sequences in arbitrary base \(b\). (English) Zbl 1313.11096

Let \(q\geq 2\) be an integer and, for \(0<c<1\), put \[ F_c^q=\{x\in [0,1)\:\{q^n x\}\geq c,\;\forall ~n\geq 0\}, \] where \(\{y\}\) denotes the fractional part of the real number \(y\).
Inspired by previous works of J. Nilsson [Isr. J. Math. 171, 93–110 (2009; Zbl 1189.11038)] and Y. B. Pesin [Dimension theory in dynamical systems: contemporary views and applications. Chicago: Univ. Chicago Press (1997; Zbl 0895.58033)], the author determines the Hausdorff dimension of \(F_c^q\). From the abstract: This dimension “can be calculated using the spectral radius of the transition matrix of the corresponding subshift”.

MSC:

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