Nilsequence of Bounded Complexity

• The paper demonstrates that any 1-bounded function on a finite abelian group with high Gowers norm correlates with a nilsequence constructed from filtered nilmanifolds.
• It uses explicit quantitative bounds on degree, rank, and Lipschitz norms to control the complexity of the nilsequence and ensure effective correlation.
• The strategy resolves the Jamneshan–Tao conjecture for bounded rank groups, extending inverse theorems in higher-order Fourier analysis and additive combinatorics.

A nilsequence of bounded complexity is a mathematical object arising in higher-order Fourier analysis, ergodic theory, and additive combinatorics, serving as a structured model for functions with large Gowers uniformity norm. The recent resolution of the Jamneshan–Tao conjecture for finite abelian groups of bounded rank establishes that any 1-bounded function on such a group with large Gowers norm must correlate nontrivially with a nilsequence whose complexity is quantitatively controlled in terms of the rank, the Gowers norm threshold, and the uniformity degree (Candela et al., 13 Jan 2026). The concept encapsulates the interaction between Lie-theoretic structure, polynomial mappings, and analytic properties of bounded Lipschitz functions on filtered nilmanifolds.

1. Nilsequences and Their Structural Parameters

A nilsequence is constructed from filtered nilmanifolds, polynomial sequences, and bounded Lipschitz functions. Let $(G/\Gamma,G_\bullet)$ denote a filtered nilmanifold of degree $k$, consisting of a nilpotent Lie group $G$, a discrete cocompact subgroup $\Gamma$, and a filtration $G_\bullet=(G=G_0=G_1\ge G_2\ge\cdots)$ with $[G_i,G_j]\subset G_{i+j}$ and $G_{k+1}=\{\mathrm{id}\}$, where each $\Gamma\cap G_i$ is cocompact in $G_i$. A map $g\colon Z\to G$ is polynomial (into $(G,G_\bullet)$) if for every collection of discrete derivatives $\partial_{h_1}\cdots\partial_{h_i}g(n)$, the image lies in $G_i$.

Given such a setup, and a 1-bounded Lipschitz function $F\colon G/\Gamma\to\mathbb{C}$, the composition $\psi(x)=F(g(x)\Gamma)$ is termed a nilsequence of degree $k$. The complexity of this nilsequence is encapsulated by an integer $M$, which bounds the following parameters: degree, dimension of $G$, length of a Malcev basis, height of rational structure constants, Lipschitz norm and $L^\infty$ norm of $F$, and the metric and distortion constants of the nilmanifold. Explicitly, “nilsequence of complexity $M$” abbreviates that all data $(G/\Gamma,G_\bullet,F,g)$ have parameters bounded by $M$, and $\|F\|_{\text{Lip}}\le M$ [(Candela et al., 13 Jan 2026), Section 1].

2. Inverse Theorem for Finite Abelian Groups of Bounded Rank

The primary result (Theorem 4.1) asserts that for any $k,R\in\mathbb{N}$ and $\delta>0$, there exist constants $C=C(k,R,\delta)$ and $\varepsilon=\varepsilon(k,R,\delta)>0$ with the following property: If $Z$ is any finite abelian group of rank $\leq R$, and $f\colon Z\to\mathbb{C}$ is 1-bounded with $\|f\|_{U^{k+1}}\ge\delta$, then there exists a filtered nilmanifold $(G/\Gamma,G_\bullet)$ of degree $\leq k$ and complexity $\leq C$, a polynomial map $g\in\mathrm{Hom}(D_1(Z),G_\bullet)$, and a 1-bounded Lipschitz function $F\colon G/\Gamma\to\mathbb{C}$ with $\|F\|_{\text{Lip}}\leq C$, such that the nilsequence $\psi(x)=F(g(x)\Gamma)$ satisfies

$$\Bigl|\mathbb{E}_{x\in Z}f(x)\overline{\psi(x)}\Bigr| \geq \varepsilon.$$

This characterizes functions with non-negligible Gowers norm as precisely those that correlate with nilsequences of explicitly bounded complexity.

3. Quantitative Control of Complexity and Correlation

The complexity bound $M$ and correlation threshold $\varepsilon$ arise through an explicit composition of effective dependencies. Leveraging the general regularity plus inverse theorem for compact nilspaces (Candela–Szegedy, see Theorem 5.2 in [CSinverse]), the approach first exhibits, for every $k,\delta$, an effective bound $M_0(k,\delta)$ such that if $\|f\|_{U^{k+1}}\ge\delta$, then $f$ correlates with a nilspace polynomial of complexity at most $M_0(k,\delta)$, with correlation at least $\delta^{2^{k+1}/2}$. For groups of bounded rank $\leq R$, the nilspace can then be chosen quasitoral with complexity $\leq M_1(k,R,\delta)$, and the subgroup and extension procedures only increase complexity by explicit functions of $(k,R,\delta)$. The resulting bounds take the form

$$M(k,R,\delta)\leq \left(\cdots((M_0(k,\delta))^{O(1,R)})\cdots\right), \qquad \varepsilon(k,R,\delta)\gg\delta^{2^{k+1}/2}/O_{k,R}(1)$$

where all dependencies are effective and explicitly trackable, though the precise closed-form for $M$ is not provided. These bounds are uniform in $R$ and, in principle, can be unwound to yield quasipolynomial or tower-type expressions in $\delta^{-1}$ (Candela et al., 13 Jan 2026).

4. Proof Outline: From Nilspace Polynomials to Global Nilsequences

The proof proceeds in two principal stages. First, by applying the Candela–Szegedy theorem, one obtains a correlation with a nilspace polynomial $F\circ\varphi\colon D_1(Z)\to\mathbb{C}$, where $\varphi$ is a $b$-balanced morphism into a compact finite-rank nilspace $X$. An inductive “quasitoral” reduction demonstrates that for rank $\leq R$ and sufficiently small $b$, $X$ must be a disjoint union of toral nilmanifolds. Key combinatorial and topological arguments (Propositions 3.5, 3.8; Lemmas 3.9, 3.11) are deployed to show that the only possible structure consistent with the bounded rank is quasitoral.

On an appropriate toral component, after possibly passing to a subgroup of bounded index $Z'\le Z$, this leads to a filtered nilmanifold $(G_0/\Gamma_0,G_{0,\bullet})$ with a 1-bounded Lipschitz nilsequence $x\mapsto F_0(g_0(x)\Gamma_0)$ on $Z'$ and nontrivial correlation with a shift of $f$ (Proposition 3.10).

Second, the nilsequence defined on $Z'$ is extended to $Z$ via a sequence of group extensions. One constructs a chain of subgroups $Z'=Z_0\le Z_1\le\dots\le Z_t=Z$, where each $Z_i\to Z_{i+1}$ is either a split extension by $\mathbb{Z}/p$ or a cyclic extension. In the split case, polynomial maps are composed with natural projections; in the cyclic case, the nilmanifold is enlarged (using semidirect products) and new one-parameter subgroups are constructed to “re-linearize” the polynomial structure for extension (Lemma 4.5, Proposition 4.8, Corollary 4.9). Each extension increases complexity by an explicit, controlled amount, culminating in a global nilsequence of required complexity (Candela et al., 13 Jan 2026).

5. Significance and Resolution of the Jamneshan–Tao Conjecture

The results confirm the Jamneshan–Tao conjecture for finite abelian groups of bounded rank, thereby elevating the role of nilsequences of bounded complexity as universal characteristic factors for functions with large Gowers uniformity norm on such groups. This theorem extends the scope of inverse Gowers norm theorems, where previously only special classes (such as cyclic groups of prime order or bounded exponent) had explicit nilsequence structure theorems.

The technical apparatus developed — including the quasitoral reduction, subgroup passage, and nilmanifold extension protocols — provides a template for understanding higher-order uniformity and structure in settings with controlled group-theoretic complexity. The result underpins further advances in additive combinatorics, ergodic theory, and theoretical computer science where analysis on finite abelian groups intersects with algebraic and geometric structure theory (Candela et al., 13 Jan 2026).

6. Connections to Related Work and Methodologies

The central regularity plus inverse results for compact nilspaces are drawn from Candela–Szegedy [CSinverse, Theorem 5.2], establishing that nilspace polynomials serve as universal obstructions to higher-order uniformity norms. The current arguments extend these foundations to a finer, group-theoretic regime (bounded rank), deploying novel combinatorial-topological machinery for the quasitoral and extension arguments (Theorem 3.1; Propositions 3.5, 3.8; Lemmas 3.9, 3.11).

The explicit tracking of complexity at all stages distinguishes this approach, guaranteeing that the structured objects (nilsequences) remain within quantifiable parameter regimes, a property critical for applications in quantitative combinatorics and theoretical computer science.

Table: Nilsequence Data and Complexity Parameters

Object	Definition/Role	Complexity Parameterized By
Filtered nilmanifold $(G/\Gamma,G_\bullet)$	Structured phase space for sequence	Degree, dimension, Malcev basis length, rational structure heights, metric data ($\leq M$)
Polynomial map $g$	Group morphism encoding polynomiality	Complexity of filtration and group
Bounded Lipschitz function $F$	Test function on nilmanifold	$\|F\|_{\text{Lip}}\leq M$, $\|F\|_\infty\leq 1$
Nilsequence $\psi(x)$	$F(g(x)\Gamma)$—model for structured functions	All above, aggregated in $M$

The critical parameters controlled uniformly are the degree $k$, the group rank $R$, and the Gowers norm lower bound $\delta$. All procedures maintain explicit bounds in terms of these variables, ensuring that the nilsequence reflects the arithmetic and algebraic structure of $f$ with effective quantitative dependence (Candela et al., 13 Jan 2026).