Discrepancy of normal numbers
نویسندگان
چکیده
منابع مشابه
On absolutely normal numbers and their discrepancy estimate
The algorithm computes the first n digits of the expansion of x in base 2 after performing triple-exponential in n mathematical operations. It is well known that for almost all real numbers x and for all integers b greater than or equal to 2, the sequence {bx}j≥0 is uniformly distributed in the unit interval, which means that its discrepancy tends to 0 as N goes to infinity. In [6], Gál and Gál...
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is called the discrepancy of (xn)n=1. In 1954 Roth (see [DrTi], [KN]) proved that for any sequence in [0, 1) limN→∞ND(N)/ log N > 0. (2) Let A be an s × s invertible matrix with integer entries. A matrix A is said to be ergodic if for almost all α ∈ R the sequence {αA}n≥1 is uniformly distributed. A vector α ∈ R is said to be normal (A normal) if the sequence {αA}n≥1 is uniformly distributed. L...
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Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a computable real number. The construction does not give the digits of the fractional expansion expl...
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is called the star-discrepancy of (z1, . . . , zN ). Here and in the sequel λ denotes the sdimensional Lebesgue measure. The Koksma-Hlawka inequality states that the difference between the integral of a function f over the s-dimensional unit cube and the arithmetic mean of the function values f(z1), . . . , f(zN ) is bounded by the product of the total variation of f (in the sense of Hardy and ...
متن کاملNormal Numbers Are Normal
A number is normal in base b if every sequence of k symbols in the letters 0, 1, . . . , b− 1 occurs in the base-b expansion of the given number with the expected frequency b−k. From an informal point of view, we can think of numbers normal in base 2 as those produced by flipping a fair coin, recording 1 for heads and 0 for tails. Normal numbers are those which are normal in every base. In this...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1986
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-47-2-175-186