Ja n 20 09 One more proof of the Erdős – Turán inequality , and an error estimate in Wigner ’ s law .

A number of proofs have appeared since then, an especially elegant one given by Ganelius [5]. In most of the proofs, the indicator of A is approximated by its convolution with an appropriate (Fejér-type) kernel. We shall present another proof, based on the arguments developed by Chebyshev, Markov, and Stieltjes to prove the Central Limit Theorem (see Akhiezer [1, Ch. 3]). In this approach, the indicator of A is approximated from above and from below by certain interpolation polynomials. The argument does not use the group structure on T, and thus works in a more general setting.