An estimate of the \(L_1\)-norm of an exponential sum (Q1290822)

The following theorem is proved. Suppose that \(A\geq 1/2\), \(1<\beta< 3/2\), and \(f(n)= [e^{A(\log n)^\beta}]\). Then for any coefficients \(\alpha_n= \pm 1\) and for \(N\geq N_1(\beta,A)>0\) the inequality \[ \int_0^1 \Biggl| \sum_{n=1}^N \alpha_n \exp(2\pi ixf(n)) \Biggr| dx\geq \exp(2^{-15} A^{-2} (\log N)^{3-2\beta}) \] is valid. This improves an estimate of S. V. Bochkarev. The proof uses an estimate of a trigonometric sum due to O. V. Popov.