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On a problem of Sidon for polynomials over finite fields

Wentang KuoShuntaro Yamagishi — 2016

Acta Arithmetica

Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, l i m n r ( ω ) / n ϵ = 0 . P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in q [ T ] , where q is a finite field of q elements....

Sidon basis in polynomial rings over finite fields

Wentang KuoShuntaro Yamagishi — 2021

Czechoslovak Mathematical Journal

Let 𝔽 q [ t ] denote the polynomial ring over 𝔽 q , the finite field of q elements. Suppose the characteristic of 𝔽 q is not 2 or 3 . We prove that there exist infinitely many N such that the set { f 𝔽 q [ t ] : deg f < N } contains a Sidon set which is an additive basis of order 3 .

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

Kathryn E. HareShuntaro Yamagishi — 2014

Acta Arithmetica

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define r N ( m ) ( ω ) to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that r N ( m ) ( ω ) < K for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

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