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On semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is semiregular if there exists a positive integer d such that each vertex of the digraph has indegree d or 0. Generalizing earlier results of the authors for the case in which k = 2 , we characterize all semiregular digraphs G ( n , k ) when k ≥ 2 is arbitrary.

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

Let G be an additive finite abelian group. For every positive integer ℓ, let d i s c ℓ ( G ) be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine d i s c ℓ ( G ) for certain finite groups, including cyclic groups, the groups G = C ₂ ⊕ C 2 m and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

Let q [ t ] be the polynomial ring over the finite field q , and let N be the subset of q [ t ] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A ⊆ N for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D ( N ) ≪ q N ( l o g N ) 7 / N .

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