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\(s(N)\)-uniform distribution modulo 1. (English) Zbl 0826.11034

The authors study the new concept of \(s(N)\)-uniform distribution. That is: For any sequence \(\underline {x}= (x_n)_{n\geq 1}\) in \([0,1)\) set \(\underline {x}^{(s)}= (x_n^{(s)} )_{n\geq 1}\), where \(x_n^{(s)}= (x_n, x_{n+1}, \dots, x_{n+s-1})\in [0,1)^s\). Let \[ \Delta (N,s, \underline {x}, Q)= \bigl|{\textstyle {1\over N}} |\{n \leq N\mid x_n^{(s)}\in Q\}|- \lambda^{(s)} (Q) \bigr|, \] where \(Q\leq [0, 1)^s\) and \(\lambda^{(s)}\) is the \(s\)- dimensional Lebesgue measure. Let \(s(N)\) be increasing monotonically and let \[ D(N, s(N), \underline {x}):= \sup_Q \Delta (N, s(N), \underline {x}, Q) \] where the supremum is extended over all “rectangles” \(Q\) of the form \(Q= \prod_{i=1}^s [a_i, b_i)\), \(a_i, b_i\in [0,1]\), where \([a, b)= [a,1) \cup [0,b)\) for \(0\leq b< a\leq 1\). \(\underline {x}\) is called \(s(N)\)-uniformly distributed if \(\lim_{N\to \infty} D(N, s(N), \underline {x})=0\). For example it is shown: Almost all \(\underline {x}\) are \(s(N)\)-uniformly distributed if \(s(N)= o(\sqrt {N/\log N})\).

MSC:

11K06 General theory of distribution modulo \(1\)
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