Irrationality-exponent sets product-dimension bounds conjecture

For $\lambda\in[2,\infty]$, let $\mathscr{W}_\lambda$ denote the set of real numbers with the corresponding irrationality exponent, and let $n\geq 1$. The irrationality-exponent product-dimension conjecture. For any $\lambda_0,\ldots,\lambda_{n-1}\in[2,\infty]$,

$$n-1+\frac{2}{\max_{0\leq i\leq n-1}\lambda_i}\geq\dim\left(\prod_{i=0}^{n-1}\mathscr{W}_{\lambda_i}\right)\geq\max\left\{n-1,\;2\sum_{i=1}^{n}\lambda_i^{-1}\right\}.$$

In particular, when all $\lambda_i$ are sufficiently large compared with $n$,

$$\dim\left(\prod_{i=0}^{n-1}\mathscr{W}_{\lambda_i}\right)>\sum_{i=0}^{n-1}\dim(\mathscr{W}_{\lambda_i}),$$

and

$$\lim_{\max\lambda_i\to\infty}\dim\left(\prod_{i=0}^{n-1}\mathscr{W}_{\lambda_i}\right)=n-1,$$

with the limit understood as in the source theorem. The analogous bounds for restricted-rational sets are proved in the paper, while these unrestricted-rational assertions are formulated as a conjecture.