Representation of measurable functions by series with respect to Walsh subsystems

For every sequence $\{\omega(n)\}_{n\in\mathbb N}$ that tends to infinity we construct a “quasiquadratic” representation spectrum $\Lambda=\{n^2+o(\omega(n))\}_{n\in\mathbb N}$: for each almost everywhere finite measurable function $f(x)$ there exists a series in the form $\sum_{k\in\Lambda}a_kw_k(x)$ that converges almost everywhere to this function, where $\{w_k(x)\}_{k\in\mathbb N}$ is the Walsh system.
We also find representation spectra in the form $\{n^l+o(n^l)\}_{n\in\mathbb N}$, where $l\in\{2^k\}_{k\in\mathbb N}$.