On the sumsets of units in a ring of matrices over Z/mZ

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Abstract

Let $M_{n, m}:=\operatorname{Mat}_{n}(\mathbb{Z} / m \mathbb{Z})$ be the ring of matrices of $n \times n \text { over } \mathbb{Z} / m \mathbb{Z} \text { and } G_{n, m}:=G l_{n}(\mathbb{Z} / m \mathbb{Z})$ be the multiplicative group of units of M_n,m with n ⩾ 2, m ⩾ 2. In this paper, we obtain an exact formula for the number of representations of any element of M_2,m as the sum of k units in M_2,m. Furthermore, by using the technique of Fourier transformation, we also give a formula for the case n ≥ 3 and m = p is a prime.

Keywords

rings of matrices / finite fields / sums of units / Fourier transformation

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