Research Problems in Function Theory — Problem 4.21

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.

Let $n\ge1$ and $a_0,\ldots,a_n\in\{-1,1\}$. For $1\le k\le n$, define the aperiodic correlations \[b_k=\sum_{j=0}^{n-k}a_{j+k}a_j. \] Put $p(z)=\sum_{j=0}^na_jz^j$. Direct expansion and Fourier orthogonality give \[|p(e^{i\theta})|^2=(n+1)+2\sum_{k=1}^nb_k\cos(k\theta),\] \[\frac1{2\pi}\int_0^{2\pi}|p(e^{i\theta})|^4\,d\theta=(n+1)^2+2\sum_{k=1}^nb_k^2. \] Thus the auxiliary constant term is $n+1$, corresponding to the $n+1$ coefficients.

Statement recorded in the supplied source.

Is there an absolute constant $A>0$ such that \[\sum_{k=1}^n|b_k|^2>An^2\] for every $n\ge1$ and every sign sequence? Such a bound would in particular imply the source's fourth-mean lower bound $\frac1{2\pi}\int_0^{2\pi}|p(e^{i\theta})|^4\,d\theta\ge n^2(1+A)$. [S2]

Short English statement

Must the sum of squared aperiodic correlations of every sign sequence be at least a fixed positive multiple of the square of its length parameter?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304021 (AMR-022-4021). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.21, its complete Update 4.21, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.