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Statement

Setting (p. 369). The signs $\varepsilon_k$, $k=1,2,\ldots$, are independent random variables taking the values $+1$ and $-1$ with probability $1/2$ each, and

$$f_n(\vartheta)=\sum_{k=1}^n\varepsilon_k\cos k\vartheta .$$

Theorem (p. 369, unnumbered). With probability $1$, for all large enough $n$,

$$\sqrt{n\log n}-4\sqrt{\frac n{\log n}}\log\log n \le\max_{0\le\vartheta\le2\pi}\lvert f_n(\vartheta)\rvert \le\sqrt{n\log n}+3\sqrt{\frac n{\log n}}\log\log n .$$

In particular $\max_{0\le\vartheta\le2\pi}\lvert f_n(\vartheta)\rvert/\sqrt{n\log n}\to1$ with probability $1$. This answers in the affirmative the question of Salem and Zygmund, who had shown that with probability $1$ the liminf of this ratio is at least $1/(2\sqrt6)$ and its limsup at most $1$, and who asked whether the ratio has a limit with probability $1$; the paper notes (p. 369) that Hayman's Research Problems in Function Theory raises the same question for power polynomials as Problem 4.17.

Remarks around the statement (p. 369). The paper proves none of the first three; the fourth is taken up on p. 377.

• The paper says the order of the error term is probably the right one and makes no attempt at best possible constants.
• For fixed $n$, it says, the $\log\log n$ can be dropped with probability $1-\varepsilon$, the constants $4$ and $3$ being replaced by a $c(\varepsilon)$, and the maximum is likely to have a limit distribution with variance a constant times $\sqrt{n/\log n}$. In its remarks on generalizations (pp. 376--377) the paper adds that the lower bound's proof rests on the values $f_n(\vartheta_m)$ at its sample points being close to independent, that this fails for the sharper fixed-$n$ result, where a denser set of points is needed as in the upper estimate, and that the same proof still applies; no proof is written out.
• The paper restricts itself to coefficients $\pm1$ and says that results can also be had for the more general coefficients Salem and Zygmund considered; p. 377 says only that the near independence fails for them too and that sharp results then need a smoother cutoff $u$.
• Before the statement, the paper says that the same result holds for power polynomials, with a minor difference in the proof given at its end; see the remark on p. 377.

Source. G. Halász, On a result of Salem and Zygmund concerning random polynomials, Studia Sci. Math. Hungar. 8 (1973), 369--377: the statement on p. 369, the proof on pp. 369--376, remarks on generalizations on pp. 376--377. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the statement and the remarks were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 369--376. Both bounds are proved first for even $n$ with failure probability $O(1/\log^4n)$, then along a sparse sequence and interpolated.

• Lower bound (pp. 370--373). A cutoff $u$ vanishes on $\lvert x\rvert\le M$ and equals $1$ for $\lvert x\rvert\ge M+\Delta$, with $\Delta=\sqrt{n/\log n}$ and $M$ of order $\sqrt{n\log n}$; $u$ is written as a Fourier--Stieltjes transform. The count $\eta=\sum_mu(f_n(\vartheta_m))$ over the $n$ points $\vartheta_m=\frac{2m-1}{2n}2\pi$ has its mean and variance estimated through the characteristic function of $f_n$, and Chebyshev's inequality gives $\max\lvert f_n\rvert\ge\sqrt{n\log n}-3.5\sqrt{n/\log n}\log\log n$ with probability $1-O(1/\log^4n)$ (p. 373).
• Upper bound (pp. 373--374). The sum over points is replaced by the integral of $u(f_n(\vartheta))$ over $(0,2\pi)$; Bernstein's inequality turns a small value of this integral into a bound on the maximum, and Markov's inequality gives $\max\lvert f_n\rvert\le\sqrt{n\log n}+2.7\sqrt{n/\log n}\log\log n$ with probability $1-O(1/\log^4n)$ (p. 374).
• All large $n$ (pp. 374--375). Borel--Cantelli along $n_j=2[e^{j^{1/3}}]$, with Lemma 4.4.1 of Salem and Zygmund bounding $\max_\vartheta\lvert f_n-f_{n_j}\rvert$ for $n_j<n\le n_{j+1}$, enlarges the constants $3.5$ and $2.7$ to $4$ and $3$.
• The cutoff (pp. 375--376). $u$ is built from a ten times continuously differentiable step, which gives the moment bounds on $dU$ used above.

Dependencies

Lemma 4.4.1 of R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245--301, cited on p. 374; Bernstein's inequality for trigonometric polynomials.

Bears on

• Problem 523: the problem asks about the power polynomial $\sum_{0\le k\le n}\epsilon_kz^k$ on $\lvert z\rvert=1$, not the cosine polynomial. Since $f_n$ is the real part of $\sum_{k=1}^n\varepsilon_ke^{ik\vartheta}$, the theorem's lower bound gives the problem's maximum at least $(1+o(1))\sqrt{n\log n}$ almost surely. The matching upper bound is the paper's extension to power polynomials, stated on its p. 377 page.