On the Erdős-Turán inequality on uniform distribution. I
The fundamental inequality of Erdős and Turán is a quantitative version of the famous Weyl criterion for uniform distribution mod 1. In the present note the author introduces a new general version of the Erdős-Turán inequality. The main result of the paper is the following Theorem. Let a function f satisfy the one-sided Lipschitz condition on [0,1] with constant L, and let \(f(0)=f(1)\). Then for any positive integer m, we have \[ [f] < \frac{4L}{m+1}+ \frac{4}{\pi} \sum^{m}_{h=1} (\frac{1}{h}- \frac{1}{m+1})| \hat f(h)|, \] where \(\hat f(h)\) is the Fourier-Stieltjes transform of f, and where [f] represents the oscillation of f.
scientific article; zbMATH DE number 4042022