Extension of the main lower bound to non-pre-commuting matrices

Establish an analogue of Theorem 1 for sums of integer linear transformations that are not necessarily pre-commuting, thereby extending the asymptotic lower bound for sums of algebraic dilates to general irreducible and coprime matrix families.

Background

The paper proves an asymptotically sharp lower bound for sums of linear transformations when the integer matrices are pre-commuting, irreducible, and coprime. Pre-commutation enables the authors to reduce the matrix problem to sums of algebraic dilates and ultimately to apply a simultaneous diagonalisation argument.

The authors identify the extension to matrices that are not necessarily pre-commuting as the main unresolved direction. They give a concrete irreducible and coprime triple of skew-symmetric matrices in dimension three as a test case, for which the methods developed in the paper do not apply.