On an Inequality of Erdimages and Turán Concerning Uniform Distribution Modulo One, II Original Research Article

A famous inequality of Erdimages and Turán estimates the discrepancy Δ of a finite sequence of real numbers by the quantity B = minKK−1 + ∑K − 1k = 1 αk/k, where the αk are the Fourier coefficients. We investigate how bad this estimate can be. We prove that in the worst case Δ is of order B3/2.