On the mean modulus of trigonometric polynomials and a conjecture of S. Chowla

Abstract:

Let $\{ {m_k}\}$ be a strictly increasing sequence of positive integers. S. Chowla (1965) conjectured that \[ \min \limits _{0 \leqq x < 1} \sum \limits _{k = 1}^n {\cos 2\pi {m_k}x < - c{n^{1/2}},} \] $c > 0$ being an absolute constant. Let ${K_1},{K_2}, \cdots ,{K_N}$ be the distinct integers ${m_l} - {m_k},1 \leqq k < l \leqq n;{r_j},1 \leqq j \leqq N$, the number of pairs (k, l) with $1 \leqq k < l \leqq n$ and ${m_l} - {m_k} = {K_j}$; and \[ r(n) = \max \limits _{1 \leqq j \leqq N} {r_j}.\] Lower bounds for $\smallint _0^1 {|\sum \nolimits _{k = 1}^n {{c_k}{e^{2\pi i{m_k}x}}} |dx,{c_k}}$ arbitrary complex numbers and $\smallint _0^1 {|\sum \nolimits _{k = 1}^n {{\gamma _k}\cos 2\pi ({m_k}x + {\alpha _k})|dx,{\gamma _k} \geqq 0,{\alpha _k}} }$ real, are obtained in terms of $n, r(n)$ and the ${c_k}$ and ${\gamma _k}$ respectively and it has been deduced that in case $r(n) = \delta$, independent of n, then \[ \min \limits _{0 \leqq x < 1} \sum \limits _{k = 1}^n {\cos 2\pi {m_k}x < - \frac {1}{{{2^{5/2}}}}\frac {1}{{{{(\delta + 1)}^{1/2}}}}{n^{1/2}}.} \]