Quantitative equidistribution of affine random walks on the torus beyond finite support and strong irreducibility

An affine random walk on the torus applies, at each step, a random toral automorphism chosen from SL_d(Z) together with a random translation. He, Lakrec and Lindenstrauss (IMRN 2021) proved a complete qualitative and quantitative equidistribution theory for such walks -- the walk's n-step distribution either tends to Haar measure, or the data are close to data trapped in a finite orbit of controlled height, with an exponential rate just below the top Lyapunov exponent of the linear part -- but under two structural restrictions: the driving measure must be finitely supported, and the group generated by the linear parts must act strongly irreducibly (with auxiliary estimates excluding finitely many primes). The problem is to remove these restrictions: to extend the dichotomy and the quantitative estimates to driving measures of finite exponential moment with infinite support and without Diophantine conditions on the translation parts, to linear parts with merely semisimple Zariski closure, to estimates uniform in the auxiliary primes, and to characterize the optimal rate and constants.

Background

Let $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ be the $d$-dimensional torus. An affine random walk on $\mathbb{T}^d$ is the Markov chain $X_{n+1}=\gamma_{n+1}X_n+u_{n+1}\ (\mathrm{mod}\ 1)$, where the pairs $(\gamma_i,u_i)$, with $\gamma_i\in\mathrm{SL}_d(\mathbb{Z})$ and $u_i\in\mathbb{T}^d$, are independent and identically distributed with common law $\mu$, a probability measure on the semidirect product $\mathrm{SL}_d(\mathbb{Z})\ltimes\mathbb{T}^d$ (the group of affine transformations $x\mapsto\gamma x+u$ of the torus). The law of $X_n$ started at $x$ is the $n$-fold convolution power $\mu^{*n}*\delta_x$. The basic question is whether $\mu^{*n}*\delta_x$ converges, in the weak-$*$ topology, to the normalized Haar (Lebesgue) measure $m_{\mathbb{T}^d}$, and at what speed; by Weyl's equidistribution criterion this is governed by the decay of the Fourier coefficients $\widehat{\mu^{*n}*\delta_x}(a)=\int e^{2\pi i\langle a,y\rangle}\,d(\mu^{*n}*\delta_x)(y)$ for nonzero integer frequencies $a\in\mathbb{Z}^d\setminus\{0\}$.

For the linear case, where $\mu$ is supported on $\mathrm{SL}_d(\mathbb{Z})\times\{0\}$, Bourgain, Furman, Lindenstrauss and Mozes (JAMS 2011) proved a quantitative equidistribution theorem: if the group $\Gamma$ generated by the support acts strongly irreducibly on $\mathbb{R}^d$ (that is, preserves no nonempty finite union of proper real linear subspaces) and contains a proximal element (an element of $\mathrm{SL}_d(\mathbb{R})$ with a unique eigenvalue of maximal modulus, i.e. a simple dominant eigenvalue), and if $\mu$ has a finite exponential moment (meaning $\int\|g\|^{\alpha}\,d\mu(g)<\infty$ for some $\alpha>0$), then from every irrational starting point the walk equidistributes at an explicit exponential rate, measured by the distance of the starting point to rational points of bounded denominator. He and de Saxcé (Duke Math. J. 2022) extended this by removing the proximality assumption, assuming instead that the Zariski closure of $\Gamma$ (the smallest algebraic subgroup of $\mathrm{SL}_d(\mathbb{R})$ over $\mathbb{C}$ containing $\Gamma$) is connected; their subsequent work (Ergodic Theory Dynam. Systems 2025) went further and handled the linear walk whenever the Zariski closure is semisimple (an algebraic group with no nontrivial connected solvable normal subgroup), with no irreducibility or proximality assumption at all, classifying the obstructions in terms of invariant subspaces and compact factors and allowing any measure with a finite exponential moment. In parallel, Benoist and Quint (Ann. of Math. 2011, 2013) developed the general theory of stationary measures (probability measures invariant in law under the walk) on homogeneous spaces, which for the affine walk yields convergence of the Cesàro averages $\frac{1}{n}\sum_{k\le n}\mu^{*k}*\delta_x$ but not of the convolution powers themselves, and gives no rate.

For genuinely affine walks, before the source paper only a result of Boyer (arXiv 2017) was available, in which the measure need not be finitely supported but the translation parts must satisfy an effective shadowing condition, a Diophantine-type hypothesis on the translation coefficients (for instance, a single translation vector $b_0$ with $d(qb_0,0)\gg q^{-C}$ for all integers $q$). The source paper of He, Lakrec and Lindenstrauss (IMRN, published online 2021) proved, for finitely supported $\mu$ on $\mathrm{SL}_d(\mathbb{Z})\ltimes\mathbb{T}^d$ whose linear part generates a strongly irreducible group satisfying the proximal-or-Zariski-connected alternative: (Theorem 1.2) the walk started at any $x$ either equidistributes to Haar measure or is trapped in a finite orbit of the group generated by the support; and (Theorem 1.3) a quantitative refinement: if at time $n$ some Fourier coefficient exceeds $t>0$, then the data $(u,x)$ must lie within distance $e^{-\lambda n}$ of data $(v,y)$ whose walk is trapped in a finite orbit of height at most $(\|a\|/t)^{C}$ (the height being the smallest denominator $q$ such that all orbit increments land in $\frac{1}{q}\mathbb{Z}^d/\mathbb{Z}^d$), for any rate $\lambda$ below the top Lyapunov exponent $\lambda_{1,\mu_0}$ of the linear part $\mu_0$ -- the almost-sure exponential growth rate of the norm of the random matrix product, which is positive by a theorem of Furstenberg under strong irreducibility. Crucially, the constants depend only on the linear part and not on the translation parts. Proposition 1.4 of the same paper shows the rate cannot be pushed beyond $\lambda_{1,\mu_0}$: near trapped data the Fourier coefficients cannot decay faster than at rate $\lambda_{1,\mu_0}$. The proof reduces certain non-concentration questions to affine walks over finite fields $\mathbb{F}_p^d$, using the spectral gap for affine groups over prime fields of Lindenstrauss and Varjú and the expansion theorem for perfect groups of Salehi Golsefidy and Varjú; both inputs hold only after excluding finitely many primes depending on the group. In an appendix to their subsequent nilmanifold paper (2022), the same three authors removed the Zariski-connectedness (and with it the proximal/connected alternative) for the torus, so the linear and affine torus theorems now hold under strong irreducibility alone. What remains open, and what this problem targets, is the rest of the gap between the affine theory and the linear theory: infinite support without Diophantine assumptions, non-strongly-irreducible (semisimple) linear parts, prime-uniform auxiliary estimates, and the sharpness of the rate.

Problem Statement

Establish the qualitative dichotomy and a quantitative equidistribution theory for affine random walks on the torus $\mathbb{T}^d$, $d\ge 2$, driven by a probability measure $\mu$ on $\mathrm{SL}_d(\mathbb{Z})\ltimes\mathbb{T}^d$, in the regimes left open by the work of He, Lakrec and Lindenstrauss and not resolved by subsequent literature. The components share one scientific objective: to bring the affine theory on the torus to the generality and precision of the existing linear theory, with the standing conclusion that unless the data (translation parts and starting point) are close to data trapped in a finite orbit (or, in the reducible setting, to the corresponding invariant rational structures), the convolution powers $\mu^{*n}*\delta_x$ equidistribute to Haar measure at an exponential rate governed by the linear part alone. Specifically:

(a) Remove the finite-support hypothesis: prove, for measures $\mu$ that are not finitely supported but have a finite exponential moment (which for this compact translation setting amounts to a finite exponential moment of the linear part), the analogue of the source paper's Theorems 1.2 and 1.3 -- equidistribution of $\mu^{*n}*\delta_x$ unless trapped in a finite orbit, with a quantitative rate and constants depending only on the linear part -- without imposing Diophantine-type conditions (such as Boyer's effective shadowing condition) on the translation parts; or show by explicit construction which additional hypothesis is necessary.

(b) Remove strong irreducibility: prove the affine analogue of He-de Saxcé's semisimple linear theorem, where the Zariski closure of the group generated by the linear parts is merely assumed semisimple (possibly acting reducibly, with compact factors), identifying the correct class of obstructing structures (finite orbits, rational translates of invariant subtori arising from the invariant decomposition and compact factors) and obtaining a quantitative rate in terms of the relevant Lyapunov exponents, uniformly in the translation parts.

(c) Make the auxiliary finite-field inputs uniform: in the mod-$p$ non-concentration estimates underlying the above (via strong approximation for algebraic groups and expansion of perfect groups), remove or quantify the dependence on the finitely many excluded primes, so that the final constants do not depend on an exceptional prime set attached to the linear group.

(d) Characterize the optimal rate: determine whether the exponential rate in the quantitative estimate can attain, or approach with controlled constants, the barrier $\lambda_{1,\mu_0}$ given by the top Lyapunov exponent of the linear part (the source paper proves every $\lambda<\lambda_{1,\mu_0}$ is achievable and Proposition 1.4 shows no $\lambda>\lambda_{1,\mu_0}$ is possible uniformly), and characterize the optimal dependence of the constants on the linear measure $\mu_0$.

For context and delimitation: the earlier assumption that $\Gamma$ contain a proximal element or have Zariski-connected closure has already been eliminated (Appendix B of the nilmanifold paper of the same authors), so the problem is stated under strong irreducibility or its semisimple relaxation, and any answer must keep the conclusions uniform in the translation parts, as the source theorems are. Since a strongly irreducible subgroup of SL_d(Z) has semisimple Zariski closure, component (a) is the strongly irreducible special case of component (b)'s semisimple class.

References

1. W. He, T. Lakrec, E. Lindenstrauss, 'Affine Random Walks on the Torus', International Mathematics Research Notices 2022(11):8003-8037 (published online 9 Jan 2021), DOI: 10.1093/imrn/rnaa322, https://doi.org/10.1093/imrn/rnaa322; author preprint arXiv:2003.03743, https://arxiv.org/abs/2003.03743
2. W. He, T. Lakrec, E. Lindenstrauss, 'Equidistribution of Affine Random Walks on Some Nilmanifolds', in 'Analysis at Large' (dedicated to the life and work of Jean Bourgain), Springer, Cham, 2022, pp. 131-171, DOI: 10.1007/978-3-031-05331-3_7, https://doi.org/10.1007/978-3-031-05331-3_7; preprint arXiv:2103.06670, https://arxiv.org/abs/2103.06670 (incl. Appendix B)
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