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

100%

Hare K., Yamagishi S.

Acta Arithmetica

2014

tom 166

nr 1

55-67

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.

On a problem of Sidon for polynomials over finite fields

100%

Kuo W., Yamagishi S.

Acta Arithmetica

2016

tom 174

nr 3

239-254

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, $lim_{n→ ∞} rₙ(ω)/n^{ϵ} = 0$. 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 $𝔽_{q}[T]$, where $𝔽_{q}$ is a finite field of q elements. More precisely, let ω be a sequence in $𝔽_{q}[T]$. Given a polynomial $h ∈ 𝔽_{q}[T]$, we define $r_{h}(ω) = |{(f,g) ∈ 𝔽_{q}[T] × 𝔽_{q}[T]: f,g ∈ ω, f + g = h, deg f, deg g ≤ deg h, f ≠ g}|$. We show that there exists a sequence ω of polynomials in $𝔽_{q}[T]$ such that $deg h ≪ r_{h}(ω) ≪ deg h$ for deg h tending to infinity.

Ograniczanie wyników

2 Acta Arithmetica

2 Yamagishi S.

1 Hare K.

1 Kuo W.

1 2016

1 2014

Wyniki wyszukiwania

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

On a problem of Sidon for polynomials over finite fields