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A \(\mathbb{Z}^d\) generalization of the Davenport-Erdős construction of normal numbers. (English) Zbl 1014.11046
From the text: “We extend the Davenport and Erdős construction of normal numbers to the \(\mathbb{Z}^d\) case. … Let \(\varphi(x)= \alpha x^r+ \alpha_1 x^{r-1}+\cdots+ \alpha_{r-1}x+ \alpha_r\) \((\alpha>0\), \(r\geq 1)\) be a polynomial with integer coefficients such that \(\varphi(n)\geq 0\) \((n=1,2,\dots)\). H. Davenport and P. Erdős [Can. J. Math. 4, 58-63 (1952; Zbl 0046.04902)] generalized Champernowne’s construction and proved that the number \(.\varphi(1) \varphi(2)\dots \varphi(n)\dots\) obtained by successively concatenating the \(b\)-expansions of the numbers \(\varphi(n)\) \((n=1,2,\dots)\) is also normal.”
The authors now define a certain double sequence and show that it is rectangular normal (any block occurs as a subblock with asymptotic frequency). In the proof they use estimates of exponential sums over polynomial sequences and the Erdős-Turán inequality. A generalization to multisequences is discussed.

MSC:
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
28D15 General groups of measure-preserving transformations
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