On the discrepancy of quasi-progressions

Abstract

A quasi-progression, also known as a Beatty sequence, consists of successive multiples of a real number, with each multiple rounded down to the largest integer not exceeding it. In 1986, Beck showed that given any 2-colouring, the hypergraph of quasi-progressions contained in {0, 1, . . . , n} corresponding to almost all real numbers in (1, ∞) have discrepancy at least log* n, the inverse of the tower function. We improve the lower bound to (log n)1/4-o(1), and also show that there is some quasi-progression with discrepancy at least (1/50)n1/6. The results remain valid even if the 2-colouring is replaced by a partial colouring of positive density.