EXPANDERS ON MATRICES OVER A FINITE CHAIN RING, II.

This work is a continuation of our previous work [D. M. Ha and H. T. Ngo. Int. J. Math., 34 (2023)], studying the expanding phenomenon in matrices over a finite chain ring with large residue field. A sum-product estimate is proved. It is shown that xy+z+t is a very strong expander, and that x(y + z) and xyz + x are moderate expanders of exponent 13. These results generalize the works of Le and Nguyen (see [J. Number. Theory, 216 (2020), pp. 174--191] and [Discrete Appl. Math., 322 (2022), pp. 166--170]. The solvability in large subsets of the equation xy = z + t is obtained, providing an answer to a question raised by Gyarmati and S\'ark\"ozy [Acta Math. Hungar., 119 (2008), pp. 259--280]. The proofs use spectral graph theory and elementary divisor theory. [ABSTRACT FROM AUTHOR]