The good-pair conjecture for the Generalised Baker–Schmidt Problem

Let $l\geq 2$ and $n\geq 2l$. A pair $(l,n)$ is called good when the conditions (I) and (II) defined earlier in the paper hold for the associated matrix construction. The induced manifolds are given by maps $g$ whose coordinate functions may be quadratic forms. Good-pair conjecture. The following claims hold:

(i) If $l\geq 2$, $n\geq 2l$, and $n-l$ is even, then $(l,n)$ is a good pair.

(ii) If $l\geq 2$, $n\geq 2l$, and $n-l\geq 4$ is even, then there exists $g$ with coordinate functions $g_j$ quadratic forms such that (I) and (II) hold on $\mathbb R^{n-l}$; hence the convergence part of the Generalised Baker–Schmidt Problem holds for the induced manifolds. Obstructions show that $n\geq 2l$ and the evenness of $n-l$ are necessary in the stated range. The simplest cases that remain open are $l=3,n=9$ and $l=4,n=8$.