“…For the setting of higher dimensional matrix ring, some generalizations could be found in [13,16]. In this paper, we shall use the incidence bound in Theorem 1.2 to give other proofs of Theorem 1.3, Theorem 1.4, Theorem 1.5 and Theorem 1.6.…”
In this paper, we study a Szemerédi-Trotter type theorem in matrix rings. More precisely, let P be a set of points and L be a set of lines in M 2 (F q ) × M 2 (F q ), we haveWe also use this theorem to give other proofs for some recent sum-product type estimates due to Karabulut, Koh, Pham, Shen, and the second listed author.
“…For the setting of higher dimensional matrix ring, some generalizations could be found in [13,16]. In this paper, we shall use the incidence bound in Theorem 1.2 to give other proofs of Theorem 1.3, Theorem 1.4, Theorem 1.5 and Theorem 1.6.…”
In this paper, we study a Szemerédi-Trotter type theorem in matrix rings. More precisely, let P be a set of points and L be a set of lines in M 2 (F q ) × M 2 (F q ), we haveWe also use this theorem to give other proofs for some recent sum-product type estimates due to Karabulut, Koh, Pham, Shen, and the second listed author.
In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x(y + z), x + yz, xy + z + t are moderate expanders over the matrix ring M n (F q ).These results generalize recent results of Y.
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