ABSTRACT
A basic problem in the theory of uniformly distributed sequences is to estimate the discrepancy of a finite collection of real numbers x 1, x 2, …,xM considered modulo 1 by an expression which depends on the sums ∑ m = 1 M e ( n x m ) , n ∈ ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2128.tif"/> . Here we write e(θ) = e 2πiθ . To fix ideas let s < t < s + 1 be real numbers and define χ s , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2129.tif"/> on ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2130.tif"/> by χ s , t ( x ) = { 1 if s < x − n < t for some n ∈ ℤ , 1 / 2 if s − x ∈ ℤ or if t − x ∈ ℤ , 0 otherwise . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq2131.tif"/>
