Khintchin Sequences in Group Endomorphisms

• Khintchin sequences are defined by averaging along orbits of group endomorphisms in compact abelian groups, ensuring convergence for every integrable function.
• Key results show that under conditions like injectivity and finite-index difference maps, recurrence sets become syndetic, generalizing classical double recurrence theorems.
• Advanced analytical tools such as Fourier-tightness and characteristic factors drive the rigorous study of recurrence phenomena in ergodic theory and combinatorics.

A Khintchin sequence of group endomorphisms is a sequence associated to an action by group endomorphisms on a compact abelian group or its measurable dynamics, for which ergodic averages along this sequence converge in a strong sense for a large class of functions, often $L^p$ spaces. This notion generalizes classical concepts of Khintchin sequences, originally defined for sequences of integers acting by rotations on the circle, to the context of group actions via endomorphisms, with deep implications in ergodic theory, harmonic analysis, and combinatorics.

1. Definition and Formal Framework

Let $(G,+)$ be a compact abelian group with Haar probability measure $\mu_G$ and $\phi\in\mathrm{Epi}(G)$ a surjective endomorphism. For a strictly increasing sequence $(n_k)_{k\ge1}\subset\mathbb{N}$, define the Khintchin class: $$K_\phi\big((n_k)\big) = \left\{ f\in L^1(G): \frac{1}{N}\sum_{k=1}^N f\big(\phi^{n_k}(x)\big) \xrightarrow{N\to\infty} \int_G f\,d\mu_G \;\;\text{in measure} \right\}.$$ The sequence $(n_k)$ is called a Khintchin sequence for $\phi$ if $K_\phi((n_k)) = L^1(G)$; that is, the ergodic averages along the orbits $\phi^{n_k}(x)$ equidistribute for every integrable function. The notion extends naturally to $L^p$-Khintchin sequences: $(n_k)$ is an $L^p$–Khintchin sequence for $\phi$ if $L^p(G) \subset K_\phi((n_k))$, which has an equivalent maximal-operator characterization for $1\le p<\infty$ in terms of weak-type $(p,p)$ bounds (Fan et al., 10 Jan 2026).

For group actions on discrete abelian groups, let $\Gamma$ be a countable discrete abelian group and $\varphi,\psi\in\mathrm{End}(\Gamma)$. In an ergodic $\Gamma$-measure-preserving system $(X,\mathcal{X},\mu,(T_g)_{g\in\Gamma})$, Khintchin sets associated to $(\varphi,\psi)$ are defined by: $$S_\varepsilon(A,\varphi,\psi) = \left\{ g\in\Gamma: \mu\left(A\cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A\right) > \mu(A)^3-\varepsilon \right\},$$ with $A\in\mathcal{X}$, $\varepsilon>0$ (Ackelsberg, 2023).

2. Fundamental Theorems on Double Recurrence

A central result is a Khintchine-type double recurrence theorem for endomorphisms of countable discrete abelian groups. If $\varphi,\psi\in\mathrm{End}(\Gamma)$ with $\delta := \psi-\varphi$ injective and $\text{Im}(\delta)$ of finite index, then for all non-null $A\in\mathcal{X}$ and all $\varepsilon>0$, the set $S_\varepsilon(A,\varphi,\psi)$ is syndetic in $\Gamma$—that is, it has bounded gaps; finitely many translates cover the whole group (Ackelsberg, 2023).

This generalizes the classical single-recurrence Khintchine theorem and answers a question left open by Ackelsberg, Bergelson, and Shalom. In multidimensional settings, e.g., $\Gamma = \mathbb{Z}^d$ and $\varphi,\psi$ induced by integer matrices $M_1, M_2$ with $\det(M_2-M_1)\ne 0$, an analogous result holds: the set of $\vec{n}\in\mathbb{Z}^d$ with large triple-intersection measure

$$\mu\left(A \cap T_{M_1\vec{n}}^{-1}A \cap T_{M_2\vec{n}}^{-1}A\right) > \mu(A)^3-\varepsilon$$

is syndetic in $\mathbb{Z}^d$ (Ackelsberg, 2023).

3. Structural Hypotheses and Characteristic Factors

Key conditions for Khintchin-type behavior in group endomorphism settings are the injectivity and finite-index image of the difference map $\delta=\psi-\varphi$ in the discrete case, or more generally the finite-index property for difference subgroups generated by $aG$, $bG$, $(b-a)G$, $(a+b)G$ for multiplicative actions by integers $a, b$ (Shalom, 2021). These requirements ensure that recurrences along the endomorphic directions do not collapse into lower-dimensional dynamics and enable the application of van der Corput differencing.

Characteristic factors controlling the limiting behavior of multiple ergodic averages are essential. In the double recurrence setting, the averages are governed by the order-2 Host–Kra (Conze–Lesigne, "quasi-affine") factor $Z_2(X)$. For more elaborate patterns, as in four-point configurations, the universal characteristic factor becomes a $2$-step nilmanifold extension of the Kronecker factor, following the Host–Kra–Ziegler framework (Shalom, 2021).

4. Fourier-Tightness and Skew-Product Ergodicity

For actions on compact abelian groups, the concept of "Fourier-tightness" is central in controlling the behavior of sequences of endomorphisms. A sequence $(\phi_k)\subset {\rm Epi}(G)$ is Fourier-tight if for each nontrivial character $\chi\in\widehat{G}$, the series

$\sum_{k=1}^\infty \|\phi_k^*(\chi)\|^2 < \infty$

is uniformly bounded in $\chi$. Fourier-tightness ensures favorable spectral properties and is a sufficient condition for several recurrence and mixing phenomena (Fan et al., 10 Jan 2026).

For skew-product actions $T_\psi(\omega,x) = (\sigma(\omega), \psi(\omega)x)$ on $\Omega\times G$, if the family $\{\psi(\sigma^k\omega):k\ge0\}$ is Fourier-tight for almost every $\omega$, then the system's ergodicity (or mixing) is equivalent to that of the base $(\Omega,\sigma,\nu)$. This mechanism underpins a large class of Khintchin sequences arising as fiber orbits in skew-product systems (Fan et al., 10 Jan 2026).

5. Examples and Constructions

Concrete constructions illustrate the variety of Khintchin sequences arising in group endomorphism contexts:

• Thue–Morse and Multiplicative Bernoulli Sequences: Sequences of the form $\lambda_n = \prod_{k=1}^n \omega_k$ with $\omega_k\in\{2,3\}$ (deterministic or i.i.d.), acting as dilations on $\mathbb{T}$, produce Khintchin sequences via uniform distribution and mixing considerations.
• Substation Sequences: The Fibonacci substitution system (e.g., $\zeta:0\mapsto 01$, $1\mapsto 0$) leads to large-scale uniform distribution of orbit sequences, yielding $L^1$-Khintchin sequences for the associated group actions.
• Principal $\mathbb{Z}^d$-Actions: For $\phi\in\mathbb{Z}[z_1^{\pm1},\dots,z_d^{\pm1}]$ of positive Mahler measure, the action on $$X_f=\ker(\phi(\sigma))\subset\mathbb{T}^{\mathbb{Z}^d}$$ produces many fiber orbits that are Khintchin sequences in factor tori (Fan et al., 10 Jan 2026).

6. Implications, Structural Properties, and Open Problems

Khintchin sequences of group endomorphisms yield syndetic sets of recurrence and popular-difference sequences with bounded gaps, extending classical results to higher-rank and multi-endomorphism contexts (Ackelsberg, 2023). The correspondence principle implies that for subsets $E\subset\Gamma$ of positive upper Banach density, sets of group elements with large multiple intersection density are syndetic, generalizing Szemerédi-type density theorems.

Several open problems remain, including precise characterization of the class of Khintchin sequences for general group endomorphisms, the necessity of Fourier-tightness in ergodicity and mixing criteria, and threshold phenomena distinguishing $L^1$ versus $L^p$-Khintchin property, especially for substitution-generated sequences and actions of higher-rank groups. The question of whether every ergodic sequence with positive lower density is $L^1$-Khintchin remains open (Fan et al., 10 Jan 2026).

7. Connections and Extensions

Khintchin recurrence for endomorphism sequences connects with the broader study of multiple recurrence and nonconventional ergodic averages, encompassing work on nilmanifolds, Host–Kra factors, Mackey group analysis, and combinatorial density principles. Extensions to multidimensional groups and nilmanifolds, as well as to $L^p$-Khintchin classes, substantially broaden the landscape of possible configurations and recurrence structures, indicating robust interaction between ergodic theory, abstract harmonic analysis, and combinatorial number theory (Ackelsberg, 2023, Shalom, 2021, Fan et al., 10 Jan 2026).