Estimate of the modulus of the characteristic function of an integral random variable (Q1081190)

The author has proved two results on the bounds for the modulus of the characteristic function \(\phi\) (t) of an integer valued random variable \(\xi\). In the first result he has proved that if \(\xi\) satisfies the condition \[ \sup_{q\in {\mathbb{Z}}, q\geq 2}\max_{0\leq r\leq q-1}P(\xi =r(mod q))<1-\delta, \] where \({\mathbb{Z}}\) is the set of integers, then its characteristic function \(\phi\) (t) for \(| t| <64\pi u/\delta\) satisfies \(| \phi (t)| \geq 1-u\) and for 16\(\pi\sqrt{2/\delta} \sqrt{u}<| t| <\pi\), \(| \phi (t)| \leq 1-u.\) Secondly he has proved that if \(\xi\) satisfies the condition \[ \sup_{a,b\in R,b\geq 2}P(\xi \in L_{a,b,})<1-\delta \] where \(L_{a,b,\epsilon}=\{x:\rho (x,\Gamma_{a,b})<\epsilon \}\) and \(\Gamma_{a,b}=\{x:\) \(x=a+bk\), \(k\in {\mathbb{Z}}\}\) then for any \(u\geq 0\), its characteristic function for \(2\pi\sqrt{u}/\sqrt{\delta}<t<\pi\) satisfies \(| \phi (t)| \leq 1-u.\) The first result is essentially due to \textit{A. S. Fomin} [Mat. Zametki 28, 791-799 (1980; Zbl 0452.60030)]. But the author has eliminated some extraneous features in the proof. The second result is the direct consequence of the estimate for a trigonometric sum which is due to \textit{D. A. Moskvin}, \textit{L. P. Postnikova} and \textit{A. A. Judin} [Teor. Veroyatn. Primen. 15, 86-96 (1970; Zbl 0217.499)].