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In this paper, we study problems about the similar configurations in F q d. Let G = (V , E) be a graph, where V = { 1 , 2 , ... , n } and E ⊆ ( V 2 ). For a set E in F q d , we say that E contains a pair of G with dilation ratio r if there exist distinct x 1 , x 2 , ... , x n ∈ E and distinct y 1 , y 2 , ... , y n ∈ E such that ‖ y i − y j ‖ = r ‖ x i − x j ‖ ≠ 0 whenever { i , j } ∈ E , where ‖ x ‖ : = x 1 2 + x 2 2 + ⋯ + x d 2 for x = (x 1 , x 2 , ... , x d) ∈ F q d. We show that if E has size at least C k q d / 2 , then E contains a pair of k -stars with dilation ratio r , and that if E has size at least C ⋅ min ⁡ { q (2 d + 1) / 3 , max ⁡ { q 3 , q d / 2 } } , then E contains a pair of 4-paths with dilation ratio r. Our method is based on enumerative combinatorics and graph theory. [ABSTRACT FROM AUTHOR]

We study some sum-product problems over matrix rings. Firstly, for A , B , C ⊆ M n (F q) , we have | A + B C | ≳ q n 2 , whenever | A | | B | | C | ≳ q 3 n 2 − n + 1 2 . Secondly, if a set A in M n (F q) satisfies | A | ≥ C (n) q n 2 − 1 for some sufficiently large C (n) , then we have max ⁡ { | A + A | , | A A | } ≳ min ⁡ { | A | 2 q n 2 − n + 1 4 , q n 2 / 3 | A | 2 / 3 }. These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra. [ABSTRACT FROM AUTHOR]