Existence of suitable ++-solutions

Establish whether there exists an infinite sequence of ++-solutions to the equation xB_{a(n)}+yB_{a(n)+1}=zD_n+vD_{n+1} whose coefficient maximum norm is bounded by HJ(a(n),n) along an increasing subsequence, where ++-solutions satisfy z+v≥1 and z,v≥0.

Background

Theorem 2 proves the classical Littlewood conjecture conditionally on the existence of infinitely many ++-solutions with a prescribed Thue–Siegel-type size bound. The subsequent discussion notes that Thue–Siegel's lemma guarantees small nonzero solutions, but does not guarantee the required sign restrictions z,v≥0 and z+v≥1. The authors explicitly leave open whether such a sequence exists, identifying this as the obstruction to applying the criterion more generally.

References

However, we do not know whether there exists an infinite sequence $\left(\overline{S}{++}(a(n_k),n_k)\right)_{n=1}{\infty}$ of type $++$-solutions satisfying n++aarettomanmonta.

n++aarettomanmonta:

1≤∥S‾++(a(nk),nk)∥∞≤H⋅J(a(nk),nk)1\le \left\| \overline{S}^{++}(a(n_k),n_k) \right\|_{\infty} \le H\cdot J(a(n_k),n_k)

— Simultaneous approximation to pairs of real numbers  (2608.20028 - Matala-aho, 20 Aug 2026) in Section 5.1, immediately following the proof of Theorem 2