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Claim. The answer to Problem 230 is no, with a power saving in the relative error. Theorem 4 of Bombieri and Bourgain states that for every $\varepsilon>0$ and every $n\ge1$ there is a polynomial $P$ with frequencies in $[0,n]$ and coefficients of modulus one such that, uniformly in $\theta$,

$$\lvert P(\theta)\rvert=\sqrt n+O\!\left(n^{1/2-1/9+\varepsilon}\right) =\sqrt n+O\!\left(n^{7/18+\varepsilon}\right),$$

where Remark 5 says the $n^\varepsilon$ may be replaced by a power of $\log n$; Theorem 7 makes the construction effective, with the same conclusion as Theorem 4 (error $n^{7/18+\varepsilon}$) and coefficients given by elementary expressions in Legendre and Jacobi symbols, and Remark 8 refines the effective version's $n^\varepsilon$ to $\exp(c\log n/\log\log n)$, short of the power of $\log n$ available for Theorem 4. The relative error $O(n^{-1/9+\varepsilon})$ tends to zero, so for every $c>0$ and every large $n$ the maximum of $\lvert P\rvert$ on the circle is below $(1+c)\sqrt n$, which refutes the constant asked for in Problem 230. The paper's polynomial has $n+1$ coefficients where the site's has $n$; replacing $\sqrt n$ by $\sqrt{n+1}$ changes the ratio by $1+O(1/n)$, and a shift of the index range changes nothing on the circle. The result sharpens Kahane's relative error $O(n^{-1/17}\sqrt{\log n})$ on the accepted Kahane page and is the quantitative resolution the site's commentary names. The card bombieri_2009_kahane_ultraflat_polynomials digests the statements (printed pp. 628--630), the construction through a smoothed quadratic-phase core, a partition of unity on an auxiliary scale, a blockwise Bernoulli construction and the Körner correction, and the derandomization through modified Jacobi-symbol sequences with Weil and Deligne bounds. The coefficients remain general points of the unit circle: the auxiliary signs select randomizing choices and do not become the coefficients, so the paper does not touch the real-sign question of Problem 1150, as the card explains.

Depends on. Nothing in this wiki: the construction is self-contained in the paper.

Acceptance. A refereed sharpening of Kahane's disproof, named by the site's curator. Refereed: E. Bombieri and J. Bourgain, On Kahane's ultraflat polynomials, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 627--703, received 3 September 2008, published 30 June 2009. Reviewed: the site's curator, Thomas Bloom, names the paper in the problem's commentary as the sharpening of Kahane's construction to a polynomial within $O(n^{7/18}(\log n)^{O(1)})$ of $\sqrt n$ on the whole circle (page last edited 23 January 2026). The card digests the paper; its arguments are not independently rederived. No formalization of this result is known here.