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


Author: J. Chidambaraswamy
Journal: Proc. Amer. Math. Soc. 36 (1972), 195-200
MSC: Primary 10L99; Secondary 42A44
DOI: https://doi.org/10.1090/S0002-9939-1972-0308075-8
MathSciNet review: 0308075
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Abstract: Let $ \{ {m_k}\} $ be a strictly increasing sequence of positive integers. S. Chowla (1965) conjectured that

$\displaystyle \mathop {\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

$\displaystyle r(n) = \mathop {\max }\limits_{1 \leqq j \leqq N} {r_j}.$

Lower bounds for $ \smallint_0^1 {\vert\sum\nolimits_{k = 1}^n {{c_k}{e^{2\pi i{m_k}x}}} \vert dx,{c_k}} $ arbitrary complex numbers and $ \smallint_0^1 {\vert\sum\nolimits_{k = 1}^n {{\gamma _k}\cos 2\pi ({m_k}x + {\alpha _k})\vert 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

$\displaystyle \mathop {\min }\limits_{0 \leqq x < 1} \sum\limits_{k = 1}^n {\co... ... {m_k}x < - \frac{1}{{{2^{5/2}}}}\frac{1}{{{{(\delta + 1)}^{1/2}}}}{n^{1/2}}.} $


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308075-8
Keywords: Trigonometric polynomial, mean modulus, Hölder inequality
Article copyright: © Copyright 1972 American Mathematical Society