Eightfold lower bound for a specific non-pre-commuting matrix triple

Prove that for the explicitly defined matrices $L_0,L_1,L_2\in\operatorname{Mat}_3(\mathbb{Z})$, one has $|L_0A+L_1A+L_2A|\geq 8|A|-o(|A|)$ for every finite set $A\subset\mathbb{Z}^3$.

Background

The authors propose a specific three-dimensional example consisting of three integer matrices that are irreducible and coprime but not pre-commuting. They state that the box construction suggests the constant 8 is asymptotically optimal.

This conjectured inequality is a concrete instance of the broader open problem of extending the paper’s main theorem beyond pre-commuting families. The authors explicitly say that they believe the bound, but do not establish it.

References

We believe that

|L_0 A+L_1 A+L_2 A|\geq 8|A|-o(|A|)

for all finite $A \subset \mathbb{Z}3$, with the box $[0, N)3$ showing that this would be asymptotically best possible.

— Sums of algebraic dilates  (2508.18586 - Conlon et al., 26 Aug 2025) in Section 8, Concluding remarks, subsection “An interesting example”