Equidistribution from approximate self-similar rescaling

Establish that every probability measure on \(\mathbb{R}^d\) whose expanding translates \(\eta_\xi^t\) are close to the approximating measures \(\eta_\xi^{t,n}\) defined in equation (\ref{close-to}) satisfies single and double equidistribution, and consequently Khintchine dichotomy.

Background

The paper proves Khintchine dichotomy for push-forwards of self-similar measures under uniformly C1+αC^{1+\alpha} maps. Its proof approximates the expanding translates associated with the nonlinear image measure by objects related to almost random walks, and then uses effective recurrence and equidistribution. The authors suggest that the dimension-bootstrap argument underlying the cited self-similar result might apply much more broadly whenever an analogous approximation property is available.

The conjecture asks whether this approximation structure alone is sufficient to obtain both single and double equidistribution and hence the Diophantine Khintchine dichotomy. It is intended as a possible route beyond images of self-similar measures.

References

So we make the following conjecture:

Suppose a probablity measure \xi on Rd satisfies that \eta_{\xi}t is close to' some \eta_\xi^{t,n} as described inclose-to`. Then such \xi would satisfy (double) equidistribution and thus Khintchine dichotomy.

close-to:

ηνt,n:=∫δkg′∗ησtrg∗δgdλn(g)\eta_\nu^{t,n}:=\int\delta_{k'_g}*\eta_\sigma^{tr_g}*\delta_g d\lambda_n(g)

— Khintchine dichotomy for images of self-similar measures and almost random walks  (2610.13099 - Shao, 9 Oct 2026) in Section 5, Future directions, immediately after equation (\ref{close-to})

What is the genuine condition needed for measures from repellers on $Rd$ to establish equidistribution as in \cref{main: single}? What about Khintchine dichotomy?

— Khintchine dichotomy for images of self-similar measures and almost random walks  (2610.13099 - Shao, 9 Oct 2026) in Section 5, Future directions, first \texttt{problem} environment

Is the Khintchine dichotomy true for self-similar measures on general nondegenerate manifolds? If not, what condition would suffice?

— Khintchine dichotomy for images of self-similar measures and almost random walks  (2610.13099 - Shao, 9 Oct 2026) in Section 5, Future directions, second \texttt{problem} environment