If f 1 c, d, then the left hand side is the proportion of points of the sequence falling in the interval c, d, and the right hand side is exactly. The weyl criterion 7 the criterion 7, applications to special sequences 8, applications to power series 10, fejers theorem, an estimate for exponential sums 15, uniform distribution of double sequences 18, notes 21, exercises 23. This means 2 1 since indicator functions are riemannintegrable. On the uniform distribution of certain sequences 187 remark. Actually this can be deduced from the definition of uniform distribution mod one. Niederreiter, uniform distribution of sequences find, read and cite all the research you. Loglike functions and uniform distribution modulo one in. For example, the drawings of a random variable uniform over a segment will be equidistributed in the.
What links here related changes upload file special pages permanent link page information. On the range of the difference between hypothetical distribution function and pykes modified empirical distribution function brunk, h. Weak universality theorem on the approximation of analytic functions by shifts of the riemann zetafunction from a beatty sequence. The law of the iterated logarithm for empirical distribution finkelstein, helen, annals of mathematical statistics, 1971. One of the equivalents of the definition of a uniformly distributed sequence modulo 1 of real numbers is the functional definition. Uniform distribution of sequences dover books on mathematics paperback may 12, 2006 by l. On iterated weighted means of bounded sequences and uniform distribution drmota, michael, tsukuba journal of mathematics, 1991. For a function f satisfying f x olog x k, k 0, and a sequence of numbers q n n, we prove by assuming several conditions on f that the sequence. The probability density function of the continuous uniform distribution is. The theory of uniform distribution began with hermann weyls celebrated paper of 1916. Eins on the distribution of numbers modulo one pdf. Proof first note that the definition of an equidistributed sequence is equivalent to the integral criterion whenever f is the indicator function of an interval.
The distribution of leading digits and uniform distribution mod 1 diaconis, persi, annals of probability, 1977. In later decades, the theory moved beyond its roots in diophantine approximations to provide common ground for topics as diverse as number theory, probability theory, functional analysis, and topological algebra. Niederreiter, uniform distribution of sequences article pdf available in bulletin of the american mathematical society 811975 january 1975. The sequence 03c9 is called uniformly distributed modulo 1 if for each. Weak universality theorem on the approximation of analytic.
As a means for investigating sequences with respect to uniform distribution, it caused in the early years of the development of the theory of uniform distribution a strong interest in exponential sums. For a streamlined proof, see the book kuipers and niederreiter, 1974. The theory of asymptotic distribution modulo one numdam. Sometimes they are chosen to be zero, and sometimes chosen to.
Niederreiter,uniform distribution of sequences, wiley interscience, 1974. In discussing the uniform distribution of sequences mod 1, the notation of kuipers and niederreiter 1974 will be followed. Clark introduction the book uniform distribution of sequences by kuipers and niederreiter, long out of print, has recently been made available again by dover books. Uniform distribution of sequences lauwerens kuipers.
See all 3 formats and editions hide other formats and editions. Uniform distribution of sequences lauwerens kuipers, harald niederreiter the theory of uniform distribution began with hermann weyls celebrated paper of 1916 and ultimately provided common ground for topics as diverse as number theory, probability theory, functional analysis, and topological algebra. There is a continuous analogue studied especially by l. Throughout, xmeans the great est integer less than or equal to x. Library of congress cataloging in publication data.
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