Browsing University of Waterloo by Author "Spencer, Craig V."
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A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3term arithmetic progression
Liu, YuRu; Spencer, Craig V. (Springer, 20090131)Let r1, . . . , rs be nonzero integers satisfying r1 + · · · + rs = 0. Let G Z/k1Z⊕· · ·⊕Z/knZ be a finite abelian group with ki ki−1(2 ≤ i ≤ n), and suppose that (ri , k1) = 1(1 ≤ i ≤ s). Let Dr(G) denote the maximal ... 
A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3term arithmetic progression (II)
Liu, YuRu; Spencer, Craig V.; Zhao, Xiaomei (Elsevier, 201102)Let G ≃ Z/k1Z ⊕ · · · ⊕ Z/kN Z be a finite abelian group with ki ki−1 (2 ≤ i ≤ N). For a matrix Y = (ai,j) ∈ Z R×S satisfying ai,1 + · · · + ai,S = 0 (1 ≤ i ≤ R), let DY (G) denote the maximal cardinality of a set ... 
A generalization of Roth's theorem in function fields
Liu, YuRu; Spencer, Craig V. (World Scientific Publishing, 200911)Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let formula denote the subset of 𝔽q[t] containing all polynomials of degree strictly less than N. For nonzero elements r1, …, rs of 𝔽q satisfying r1 ... 
A Prime Analogue of Roth's Theorem in Function Fields
Liu, YuRu; Spencer, Craig V. (Springer New York, 2015)Abstract. Let Fq[t] denote the polynomial ring over the nite eld Fq, and let PR denote the subset of Fq[t] containing all monic irreducible polynomials of degree R. For nonzero elements r = (r1; r2; r3) of Fq satisfying ... 
Roth's theorem on systems of linear forms in function fields
Liu, YuRu; Spencer, Craig V.; Zhao, Xiaomei (Institute of Mathematics, 2010)1. Introduction. For r, s ∈ N = {1, 2, . . .} with s ≥ 2r + 1, let (bi,j ) be an r×s matrix whose elements are integers. Suppose that bi,1+· · ·+bi,s = 0 (1 ≤ i ≤ r). Suppose further that among the columns of the matrix, ...