---

Statement

Conventions as on the Theorem 7 page.

Proposition 16 (pp. 10--11). Let $p>0$ and let $f(x,y)=\sum_{k=1}^Ke(n_kx+m_ky)$ as in (16), with $K\in\mathbb N$, the $n_k$ and $m_k$ nonnegative integers and the $m_k$ strictly increasing. Suppose its marginal $p$-integral $F(x)=\int_0^1|f(x,y)|^p\,dy$ of (17) has a strict maximum at $a$, where $a=0$ or $a=1/2$. Then there is full p-concentration with gap at $a$.

The paper notes (Remark 18, p. 12) that $|f(-x,-y)|=|f(x,y)|$, so $F$ is even, and a unique maximum on $\mathbb T$ can occur only at 0 or $1/2$. It credits the use of bivariate idempotents and Riesz products to a suggestion of Terence Tao (p. 8).

Source. Aline Bonami and Szilárd Gy. Révész, Integral concentration of idempotent trigonometric polynomials with gaps, arXiv:0707.3023v2 (16 October 2008): Proposition 16 on pp. 10--11, proof pp. 11--12; Lemma 17 on p. 11. The edition is the one identified on the source card.

Read depth. Claims checked: the statement and Lemma 17 were read clause by clause on the printed pages. The proof was read but not checked step by step.

Proof pointer

Take $M$ above all $n_k,m_k$ and the Riesz product $g_{R,J}(x)=\prod_{j=1}^Jf(x,R^jx)$ of (18). For $R>M(J+1)$ it is an idempotent, and its gaps exceed any given $N$ once $R$ is large in terms of $J$, $M$ and $N$. First fix $J$ so large that $F^J$ puts all but a fraction $\varepsilon$ of its integral on $[a-\delta,a+\delta]$, which the strict maximum allows. Then Lemma 17 (p. 11), an equidistribution statement proved from the Riemann--Lebesgue lemma, shows that the integral of $|g_{R,J}|^p$ over any fixed interval tends to that of $F^J$ as $R\to\infty$.

Dependencies

Lemma 17 of the paper.

Bears on

No Erdős problem page directly.