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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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