Extremal structure of the optimal sine and sawtooth inequalities: full characterization of optimal coefficient vectors and classification of extremal measures

Esser, Tao, Totaro, and Wang (Journal of Fourier Analysis and Applications, 2022) determined the optimal inequalities $\sum_{k=1}^m a_k \sin(kx) \le 1$ and $\sum_{k=1}^m a_k g(kx) \le 1$ over nonnegative coefficients $a_k$, where $g$ is the sawtooth (signed fractional part) function: the maximal total coefficient mass is $c_m = \frac{2}{m+1}\sum_{1 \le j \le m,\, j \text{ odd}} \cot\frac{\pi j}{2m+2}$ for the sine problem and $r + 1 + m/2^r$ (with $2^r < m \le 2^{r+1}$) for the sawtooth problem, and one explicit optimal inequality and one explicit extremal probability measure are exhibited for each. The extremal structure beyond these single solutions is left undetermined: the authors state that the requirement that an optimal sawtooth inequality attain the bound 1 at every point of the support of the extremal measure does not fully determine its coefficients, that other optimal linear combinations exist, and they make no classification or uniqueness claim for the extremal measures of either problem. This problem asks for the complete extremal structure at the source's full generality: for every positive integer $m$, an explicit characterization of the set of all optimal coefficient vectors (the optimal face of the semi-infinite linear program, including its dimension and the uniqueness question for the sine problem), together with a classification of all probability measures attaining the equivalent equidistribution optima, deciding whether the exhibited extremal measures are unique. Both questions remain open.

Background

A recent extremal problem for trigonometric sums, introduced and solved in its constant-determination aspect by Esser, Tao, Totaro, and Wang (2022), asks for the best inequalities of the form $\sum_{k=1}^m a_k \sin(kx) \le 1$ and $\sum_{k=1}^m a_k g(kx) \le 1$, required to hold for all real $x$, where the coefficients $a_k$ are nonnegative real numbers and one maximizes the total mass $\sum_{k=1}^m a_k$. Here $m$ is a positive integer and $g$ denotes the sawtooth function (also called the signed fractional part), $g(x) = x + \lfloor 1/2 - x \rfloor$: the 1-periodic function with values in $(-1/2, 1/2]$ that agrees with $x$ on the interval $(-1/2, 1/2)$ and jumps down by 1 at each half-integer.

Each of these is a semi-infinite linear program: finitely many variables $a_k \ge 0$ but one linear inequality per point $x$ of the relevant circle — the circle $\mathbb{R}/\mathbb{Z}$ for the sawtooth problem, where the functions $g(kx)$ are 1-periodic, and the circle $\mathbb{R}/2\pi\mathbb{Z}$ for the sine problem, where the functions $\sin(kx)$ are $2\pi$-periodic. Its dual concerns equidistribution: writing $\mathbb{E}_\mu h = \int h \, d\mu$ for the expectation of a function $h$ with respect to a Borel probability measure $\mu$ on the real line, the maximal total mass of an inequality of the above form equals $1/\lambda_m$, where

$$\lambda_m = \max_{\mu} \, \min_{1 \le k \le m} \, \mathbb{E}_\mu \, g(kx)$$

and identically with $\sin(kx)$ in place of $g(kx)$ for the sine problem. Thus the optimal inequality encodes how small all of $g(x), g(2x), \ldots, g(mx)$ (respectively $\sin x, \ldots, \sin mx$) can simultaneously be on average — a sharp quantitative equidistribution statement on the unit circle.

Esser–Tao–Totaro–Wang determined both optima exactly. For the sine problem the optimal total mass is

$$c_m = \frac{2}{m+1} \sum_{\substack{1 \le j \le m \\ j \text{ odd}}} \cot \frac{\pi j}{2m+2} = \frac{2}{\pi} \log(m+1) + \frac{2}{\pi}\left(\gamma + \log \frac{4}{\pi}\right) + o(1),$$

where $\gamma$ is the Euler–Mascheroni constant, and an explicit optimal inequality with all $a_k \ge 0$ is produced by a discrete Fourier transform computation on the cyclic group $\mathbb{Z}/(2m+2)$. For the sawtooth problem, letting $r$ be defined by $2^r < m \le 2^{r+1}$, the optimal total mass is $r + 1 + m/2^r$ and the optimal equidistribution bound is $\lambda_m = 2^r/((r+1)2^r + m) < (\log 2)/\log m$. In each case the paper exhibits one extremal (that is, optimal) probability measure: for the sawtooth, the measure $\mu_m$ placing mass $2^r/((r+1)2^r+m)$ at each of the points $1/2, 1/4, \ldots, 1/2^{r+1}$ and mass $m/((r+1)2^r+m)$ at $1/(2m)$; for the sine, the measure placing mass $2\cot(\pi j/(2m+2))/((m+1)c_m)$ at $\pi j/(m+1)$ for each odd $j \le m$.

What the paper does not determine is the extremal structure beyond one exhibited solution of each kind. By the duality, an optimal inequality must be tight — its left-hand side must attain the bound 1 at every point of the support of an extremal measure (a complementary-slackness condition) — but the authors note (their Remark 1.3) that their sawtooth inequality was discovered by numerical experimentation guided by this tightness requirement, that the requirement does not fully determine the coefficients of the combination, and that other linear combinations of the $g(kx)$ also establish the optimal bound; when $m$ is a power of two, the paper's two displayed optimal sawtooth inequalities are already distinct, so the optimal sawtooth inequality is provably not unique. For the sine problem no analogous non-uniqueness is exhibited, but neither is uniqueness proved. On the measure side, the paper makes no claim that the exhibited $\mu_m$ is the unique extremal measure for either problem, and no classification of extremal measures is given.

Two neighboring lines of work on extremal trigonometric sums should not be confused with this question. Vaaler's inequality (1985) — described in the source as the closest relative in the literature — concerns sharp trigonometric minorants of the form $\sum_k a_k \sin(2\pi k x) \ge x - 1/2$ on $[0, 1/2]$, obtained by extremal-majorant methods, a different objective. The Rogosinski–Szegő problem, studied recently by Dmitrishin, Stokolos, and Trebels (2025), seeks extremal coefficient estimates for sine polynomials that are themselves nonnegative, whereas here the polynomial is bounded above by 1 with free nonnegative coefficients and the total mass is maximized. Neither line addresses the optimal faces or the extremizers of the present inequalities.

Problem Statement

For each positive integer $m$, determine the complete extremal structure of the two dual pairs of optimization problems solved, as to their optimal values, by Esser–Tao–Totaro–Wang. Throughout, $g$ denotes the sawtooth (signed fractional part) function $g(x) = x + \lfloor 1/2 - x \rfloor$, the 1-periodic function with values in $(-1/2, 1/2]$ that agrees with $x$ on the interval $(-1/2, 1/2)$ and jumps down by 1 at each half-integer.

(1) Optimal coefficient vectors. Characterize explicitly, for every $m$, the full set of optimal coefficient vectors in each problem: all $a = (a_1, \ldots, a_m)$ with $a_k \ge 0$ such that $\sum_{k=1}^m a_k g(kx) \le 1$ for all real $x$ (respectively $\sum_{k=1}^m a_k \sin(kx) \le 1$ for all real $x$) and the total mass $\sum_{k=1}^m a_k$ attains the optimal value $r + 1 + m/2^r$, where $r$ is the integer defined by $2^r < m \le 2^{r+1}$ for $m \ge 2$, with the convention $r = 0$ when $m = 1$ (respectively the optimal value $c_m = \frac{2}{m+1}\sum_{1 \le j \le m,\, j \text{ odd}} \cot\frac{\pi j}{2m+2}$). This set is the optimal face of the semi-infinite linear program — the set of all feasible vectors attaining the optimal value. The characterization includes determining its dimension and facial structure (the source establishes that the optimal sawtooth inequality is not unique and that tightness on the support of the extremal measure does not fully determine the coefficients), and deciding, for the sine problem, whether the optimal coefficient vector is unique.

(2) Extremal measures. Classify, for every $m$, all probability measures $\mu$ (Borel probability measures on the real line, identified modulo the relevant period: equivalently, probability measures on the circle $\mathbb{R}/\mathbb{Z}$ for the sawtooth problem, where $g(kx)$ is 1-periodic, and on the circle $\mathbb{R}/2\pi\mathbb{Z}$ for the sine problem, where $\sin(kx)$ is $2\pi$-periodic) that attain the equidistribution optimum $\lambda_m = 2^r/((r+1)2^r + m)$ for the sawtooth problem (respectively $1/c_m$ for the sine problem), with $r$ as in (1), in $\max_\mu \min_{1 \le k \le m} \mathbb{E}_\mu g(kx)$ (respectively with $\sin(kx)$), where $\mathbb{E}_\mu h = \int h \, d\mu$; and decide, for each problem, whether the measure $\mu_m$ exhibited in the source — for the sawtooth, the measure placing mass $2^r/((r+1)2^r+m)$ at each of the points $1/2, 1/4, \ldots, 1/2^{r+1}$ and mass $m/((r+1)2^r+m)$ at $1/(2m)$; for the sine, the measure placing mass $2\cot(\pi j/(2m+2))/((m+1)c_m)$ at $\pi j/(m+1)$ for each odd $j \le m$ — is the unique extremizer.

The two problems may be treated separately, but each is to be resolved in full generality: for every positive integer $m$, with no restriction on the form of the answer.

References

1. Louis Esser, Terence Tao, Burt Totaro, and Chengxi Wang, 'Optimal sine and sawtooth inequalities', Journal of Fourier Analysis and Applications 28 (2022), no. 2, article 14. DOI: 10.1007/s00041-022-09909-7; arXiv:2107.11058. https://doi.org/10.1007/s00041-022-09909-7
2. Louis Esser, Burt Totaro, and Chengxi Wang, 'Varieties of general type with doubly exponential asymptotics', Transactions of the American Mathematical Society, Series B 10 (2023), 288-309. DOI: 10.1090/btran/125; arXiv:2109.13383. https://doi.org/10.1090/btran/125
3. Luka Podrug and Dragutin Svrtan, 'Some Refinements of Formulae Involving Floor and Ceiling Functions', arXiv:2302.01900 (2023). https://arxiv.org/abs/2302.01900
4. Jeffrey D. Vaaler, 'Some extremal functions in Fourier analysis', Bulletin of the American Mathematical Society (New Series) 12 (1985), 183-216. DOI: 10.1090/s0273-0979-1985-15349-2. https://doi.org/10.1090/s0273-0979-1985-15349-2
5. Dmitriy Dmitrishin, Alexander Stokolos, and Walter Trebels, 'Extremal polynomials for the Rogosinski–Szegő estimates of the third coefficient of nonnegative sine polynomials', arXiv:2509.22238 (2025). https://arxiv.org/abs/2509.22238