Constantes d'Erdös-Turán

The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, $c_1$ and $c_2$. We show that $c_1\geq 1$ and $c_2\geq 2/\pi$, and we provide a set of admissible pairs $(c_1;c_2)$ that are numerically close to the hypothetical optimum $(1;2/\pi)$, including $(1;0.653)$ and $(1.1435;2/\pi)$.