González Robert–et al. twisted lim sup conjecture

Let ${\bm \omega}=({\bm \omega}_{n})_{n\ge 1}$ be a sequence of points in $\mathbb{R}^{d}$, and let ${\bf r}=(r_1,\ldots,r_d)$ be a $d$-tuple of non-increasing functions. For the twisted lim sup set

$$\mathcal{T}({\bm \omega},{\bf r}):=\left\{ {\bm \beta}\in[0,1)^{d}: {\bm \omega}_{n}\in {\bm \beta}+\mathcal{R}_{n}({\bf r})\pmod 1\text{ for infinitely many }n\in\mathbb{N}\right\},$$

González Robert–et al. conjecture. For almost every sequence ${\bm \omega}$ in $\mathbb{R}^{d}$, for every $d$-tuple of non-increasing functions ${\bf r}=(r_1,\ldots,r_d)$,

$$\mathcal{L}^{d}\big(\mathcal{T}({\bm \omega},{\bf r})\big)=\begin{cases}0,&\text{if }\sum_{n=1}^{\infty}\prod_{i=1}^{d}r_i(n)<\infty,\\1,&\text{if }\sum_{n=1}^{\infty}\prod_{i=1}^{d}r_i(n)=\infty. \end{cases}$$

This conjecture predicts a zero–one law for twisted lim sup sets generated by almost every sequence. The supplied text attributes it to González Robert et al. and does not provide evidence of resolution.