Let $ M_{n, m}: = Mat_n(\mathbb{Z}/m\mathbb{Z}) $ be the ring of matrices of $ n\times n $ over $ \mathbb{Z}/m\mathbb{Z} $ and $ G_{n, m}: = Gl_n(\mathbb{Z}/m\mathbb{Z}) $ be the multiplicative group of units of $ M_{n, m} $ with $ n\geqslant 2, m\geqslant 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\ge3 $ and $ m = p $ is a prime.
Citation: Yifan Luo, Kaisheng Lei, Qingzhong Ji. On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $[J]. Electronic Research Archive, 2025, 33(3): 1323-1332. doi: 10.3934/era.2025059
Let $ M_{n, m}: = Mat_n(\mathbb{Z}/m\mathbb{Z}) $ be the ring of matrices of $ n\times n $ over $ \mathbb{Z}/m\mathbb{Z} $ and $ G_{n, m}: = Gl_n(\mathbb{Z}/m\mathbb{Z}) $ be the multiplicative group of units of $ M_{n, m} $ with $ n\geqslant 2, m\geqslant 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\ge3 $ and $ m = p $ is a prime.
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Yifan Luo, Kaisheng Lei, Qingzhong Ji. On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $[J]. Electronic Research Archive, 2025, 33(3): 1323-1332. doi: 10.3934/era.2025059
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