On polynomial d-chaos via d-dissociated character subsystems on compact abelian groups

Anna Kazakova

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Pith reviewed 2026-05-10 14:43 UTC · model grok-4.3

classification 🧮 math.FA math.DS

keywords chaosespolynomialabelianchaoscompactdegreedissociatedfactors

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Polynomial d-chaoses and tetrahedral chaoses from d-dissociated character subsystems on compact abelian groups are q-lacunary and 2d/(d+1)-Sidon systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

On compact abelian groups, characters act as the fundamental waves or frequencies for Fourier analysis. A d-dissociated subsystem means the characters satisfy a strong independence condition: products of up to d of them coincide only in trivial ways. Polynomial d-chaos collects all possible products of exactly d characters from the subsystem, allowing repeats, while tetrahedral chaos requires all factors to be distinct. The result shows these collections behave like sparse (q-lacunary) sets and obey a Sidon inequality with the specific constant 2d/(d+1), meaning sums over them have controlled norms similar to orthogonal systems. This builds on classical one-dimensional results for lacunary series and Sidon sets but applies them to multiple products.

Core claim

We prove that polynomial d-chaoses (and, consequently, the tetrahedral chaoses) with respect to d-dissociated subsystems of characters on compact abelian groups are q-lacunary and 2d/(d+1)-Sidon systems.

Load-bearing premise

The character subsystems must be d-dissociated, i.e., satisfy the specific independence condition on products up to degree d; without this the lacunarity and Sidon conclusions do not follow.

read the original abstract

In this paper, we study polynomial chaoses of degree $d$ constructed from sequences of functions; that is, sets of all possible $d$-fold products of sequence elements, allowing repeated factors. The tetrahedral chaos of degree $d$ is defined as the subset consisting of products with pairwise distinct factors. We prove that polynomial $d$-chaoses (and, consequently, the tetrahedral chaoses) with respect to $d$-dissociated subsystems of characters on compact abelian groups are $q$-lacunary and $2d/(d+1)$-Sidon systems.

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Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of characters on compact abelian groups and the domain-specific definition of d-dissociated subsystems; no free parameters or new entities are introduced.

axioms (2)

standard math Characters on compact abelian groups form an orthonormal basis for L2 and satisfy standard product rules.
Fundamental background in harmonic analysis invoked for all such results.
domain assumption d-dissociated subsystems satisfy the independence condition that non-trivial products of at most d characters coincide only trivially.
This is the central hypothesis from which the lacunarity and Sidon conclusions are derived.

pith-pipeline@v0.9.0 · 5387 in / 1339 out tokens · 52177 ms · 2026-05-10T14:43:35.865346+00:00 · methodology

On polynomial d-chaos via d-dissociated character subsystems on compact abelian groups

Core claim

Load-bearing premise

discussion (0)