Research Problems in Function Theory — Problem 7.33

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For distinct real frequencies $\lambda_1,\ldots,\lambda_N$, let \[P(\theta)=\sum_{j=1}^N e^{i\lambda_j\theta},\qquad \mu(P)=\inf_{\theta\in\mathbb R}|P(\theta)|.\] The rational-frequency special case can be converted, by a common denominator and a unit-modulus factor, to a polynomial with coefficients zero or one evaluated on the unit circle.

Statement recorded in the supplied source.

Determine sharp bounds for $\mu(P)$ as the distinct frequencies vary, in terms of the number of terms $N$. In particular, improve or determine the sharpness of the recorded universal bound $\mu(P)\le\sqrt{N-1}$ for $N\ge2$, including the rational-frequency polynomial case. [S2] The update reports an explicit four-term calculation and examples with few terms and large minimum modulus. [S2]

Short English statement

Determine how large the minimum modulus of a sum of distinct unit oscillations can be for a given number of terms, including the corresponding zero-one polynomial problem.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307033 (AMR-022-7033). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.33 and its complete update. https://arxiv.org/abs/1809.07200.