Equidistributed sequence

In mathematics, a sequence {s1, s2, s3, ...} of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that interval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.This means 2 → 1 (since indicator functions are Riemann-integrable), and 1 → 2 for f being an indicator function of an interval. It remains to assume that the integral criterion holds for indicator functions and prove that it holds for general Riemann-integrable functions as well.Conversely, suppose Weyl's criterion holds. Then the Riemann integral criterion holds for functions f as above, and by linearity of the criterion, it holds for f being any trigonometric polynomial. By the Stone–Weierstrass theorem and an approximation argument, this extends to any continuous function f. In mathematics, a sequence {s1, s2, s3, ...} of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that interval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration. A sequence {s1, s2, s3, ...} of real numbers is said to be equidistributed on a non-degenerate interval if for any subinterval of we have (Here, the notation |{s1,...,sn }∩| denotes the number of elements, out of the first n elements of the sequence, that are between c and d.) For example, if a sequence is equidistributed in , since the interval occupies 1/5 of the length of the interval , as n becomes large, the proportion of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that {sn} is a sequence of random variables; rather, it is a determinate sequence of real numbers. We define the discrepancy DN for a sequence {s1, s2, s3, ...} with respect to the interval as A sequence is thus equidistributed if the discrepancy DN tends to zero as N tends to infinity. Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. For stronger criteria and for constructions of sequences that are more evenly distributed, see low-discrepancy sequence. Recall that if f is a function having a Riemann integral in the interval , then its integral is the limit of Riemann sums taken by sampling the function f in a set of points chosen from a fine partition of the interval. Therefore, if some sequence is equidistributed in , it is expected that this sequence can be used to calculate the integral of a Riemann-integrable function. This leads to the following criterion for an equidistributed sequence: Suppose {s1, s2, s3, ...} is a sequence contained in the interval . Then the following conditions are equivalent: