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Statement

Conventions as on the Theorem 7 page.

Proposition 10 (p. 6, quoted). "Full p-concentration with gap at $1/2$ holds whenever $p>0$ is not an even integer. On the other hand, for $p=2k\in2\mathbb N$, $c_{2k}(1/2)=1/2$."

The paper calls this the key to full concentration at points other than 0 (p. 6), and uses it in Proposition 12 (p. 7) and Proposition 33 (p. 19) to obtain $c_p=1$ with gaps for $p\notin2\mathbb N$.

Source. Aline Bonami and Szilárd Gy. Révész, Integral concentration of idempotent trigonometric polynomials with gaps, arXiv:0707.3023v2 (16 October 2008): Proposition 10 on p. 6; the even case on p. 10; the proof of the non-even case on pp. 12--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

Both non-even cases apply Proposition 16. For $0<p<2$ the marginal $p$-integral of $1+e(y)+e(x+2y)$ has a strict maximum at $1/2$ (Proposition 19, p. 12). For $p>2$ not an even integer and $k$ an odd integer larger than $p/2$, Proposition 21 (p. 13) shows that the marginal of $(1+e_1(x)e_k(y))(1+e_1(x)e_{k+1}(y))$ has a strict maximum at $1/2$; its Fourier coefficients come from the Mockenhaupt--Schlag expansion of $|\cos\pi y|^p$ and have the signs that put the maximum at $1/2$. For $p=2k$ the value $c_{2k}(1/2)=1/2$ is the argument of p. 10 recorded on the Theorem 7 page.

Dependencies

Proposition 16; Propositions 19 and 21 of the paper; the construction of Mockenhaupt and Schlag on the Hardy--Littlewood majorant problem; the positive definite value at $1/2$ for $p=2$, cited from Déchamps-Gondim, Lust-Piquard and Queffélec.

Bears on

No Erdős problem page directly.