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A non-linear Roth theorem for thick Cantor sets
Alex McDonald, Micah Nguyen
https://arxiv.org/abs/2509.17880 https://arxiv.org/pdf/2509.17880

A non-linear Roth theorem for thick Cantor sets
We prove that for any function $f$ satisfying certain mild conditions and any Cantor set $K$ with Newhouse thickness greater than $1$, there exists $x\in K$ and $t>0$ such that \[ \{x-t,x,x+f(t)\}\subset K. \] This is an extension of previous work on the existence of three-term arithmetic progressions in Cantor sets to the non-linear setting.

Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set
Hugo Marsan, Mathieu Sablik
https://arxiv.org/abs/2508.00461 https://

Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set
Consider the dynamical system constitued by a continuous function $F:\mathcal{A}^\mathbb{N}\to\mathcal{A}^\mathbb{N}$ where $\mathcal{A}$ is a finite alphabet. The perturbed counterpart, denoted by $F_ε$, is obtained after each iteration of $F$ by modifying each cell independently with probability $ε\in[0,1]$ and choosing the new value uniformly. We characterize the possible sets of $ε\in[0,1]$ such that $F_ε$ has a unique measure. These sets are exactly the $G_δ$ sets (countable intersect…

The centered maximal operator removes the non-concave Cantor part from the gradient
Panu Lahti, Julian Weigt
https://arxiv.org/abs/2510.01936 https://arxiv…

The centered maximal operator removes the non-concave Cantor part from the gradient
We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $M f(x)>f^*(x)$. We further deduce from this that if the variation measure of $f$ has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of $M f$ can be upgraded to Sobolev regularity.