In the specified construction, the function h reaches every Holder level below exponent one but is not Lipschitz, while the associated toral map f is a pseudo-rotation that fails bounded mean motion and C0-rigidity for every irrational alpha.
This is a mathematical proof about defined objects rather than an empirical sample. The paper asks how the lacunary skew product behaves in regularity, quantitative deviation and rigidity. Its theoretical inputs are an irrational alpha, the denominators q_j of its continued-fraction convergents, the function h and the associated skew product f.
The exact gap at the endpoint
Holder continuity is the regularity condition h satisfies at every exponent below one. Lipschitz continuity is the endpoint it does not reach. Put simply, the function can meet conditions arbitrarily close to that endpoint without meeting the endpoint itself.
The non-Lipschitz conclusion comes from a calculation with Fourier coefficients and a Riemann-Lebesgue contradiction. The proof is self-contained except for standard facts about continued fractions and Fourier series.
Rotation does not guarantee bounded mean motion
The same map f has rotation vector (alpha, 0) and is a toral pseudo-rotation. In this construction, the relevant Birkhoff sums are not uniformly bounded; bounded mean motion fails for every irrational alpha. Bounded mean motion here is the condition that quantities accumulated along repeated iterates stay uniformly bounded.
Arithmetic sets the deviation rate
The paper gives quantitative deviation results under stated Diophantine conditions on alpha. When alpha satisfies DC(tau) with tau greater than one, the deviation quantity D_n(f) obeys D_n(f) <= C n^(1 - 1/tau), and f has (C, 1 - 1/tau)-deviation. The power exponent in that bound is tied to tau.
At the borderline condition DC(1), the form changes: D_n(f) <= C log(2n). In that case, f has (Cdelta, delta)-deviation for every 0 < delta < 1, but it does not have (C, 0)-deviation.
The constants in these bounds may depend on the Diophantine constant gamma and on tau, but not on the iterate n. Thus the arithmetic assumptions can affect the constant even though n is the iterate being varied.
A separate test for rigidity
Non-rigidity is unconditional within the irrational-alpha setting: f is not C0-rigid for every irrational alpha, and the conclusion does not require a Diophantine condition. In ordinary terms, no sequence of iterates becomes uniformly close to the identity.
For sufficiently large n, the proof chooses j as the smallest index for which q_(j+1) >= 4n. That specific choice is part of the argument leading to the failure of C0-rigidity.
A tightly defined result
The findings concern the specified lacunary skew product and its function h; the analysis has a theoretical input rather than an empirical sample. The universal statements in this construction are the pseudo-rotation, bounded-mean-motion failure and non-rigidity for every irrational alpha, while the quantitative deviation statements are tied to the corresponding Diophantine conditions.
The document is an arXiv preprint, version 2, dated 27 Aug 2026.
Paper data and sources
Original title: Regularity, quantitative deviation, and non-rigidity of a lacunary skew product
Authors: Yinshan Chang, Jian Wang, Junchang Zhou
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text