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Statement

Conventions as on the Theorem 7 page: concentration at a point $a$ (Definition 1, p. 2) and with gap (Definition 6, p. 4).

Proposition 9 (p. 5). For every $p>0$ with $p\ne2$ there is full p-concentration with gap at 0: for every $\varepsilon>0$, every symmetric open set $E\ni0$ and every $N>0$ there is an idempotent $f$ with gaps larger than $N$ and $\int_E|f|^p\ge(1-\varepsilon)\int_{\mathbb T}|f|^p$. For $p=2$ there is no point $a\in\mathbb T$ at which positive concentration with arbitrarily large gaps holds.

The paper remarks (p. 5) that for $p>1$ the Dirichlet kernel already gives full concentration at 0 without the gap requirement, and cannot be used for $p\le1$; for $p>1$ other than 2 the novelty is that the peaking polynomial may have arbitrarily large gaps. It reads the case $p\ne2$ as the failure of any Ingham-type inequality outside $L^2$, answering negatively a question of Zygmund (p. 6 and Remark 15, p. 9).

Source. Aline Bonami and Szilárd Gy. Révész, Integral concentration of idempotent trigonometric polynomials with gaps, arXiv:0707.3023v2 (16 October 2008): Proposition 9 on p. 5; the case $p=2$ proved on p. 9; the case $p\ne2$ proved in Section 3, pp. 10--14. The edition is the one identified on the source card.

Read depth. Claims checked: the statement and the cited proof locations were read clause by clause on the printed pages. The proofs were read but not checked step by step.

Proof pointer

For $p\ne2$ the paper applies Proposition 16 to a bivariate idempotent whose marginal $p$-integral has a strict maximum at 0: for $p>2$ it is $1+e(y)+e(x+2y)$, whose marginal is maximal at 0 by Proposition 19 (p. 12); for $0<p<2$ it is $(1+e_1(y))(1+e_1(x)e_3(y))$, whose marginal the paper says has a strict maximum at 0 by the Mockenhaupt--Schlag computations (p. 14). For $p=2$ the argument of p. 9 bounds the share of $\int|f|^2$ on a short interval $E$ around the point by $2|E|$ plus a Fourier tail of a triangle function beyond the gap, which tends to 0; the paper writes it at 0, and the same computation applies after translation.

Dependencies

Proposition 16; Proposition 19 of the paper; the coefficient computations of Mockenhaupt and Schlag; Parseval's identity.

Bears on

No Erdős problem page directly.