On a problem of Sidon for polynomials over finite fields

Wentang Kuo, and Shuntaro Yamagishi. "On a problem of Sidon for polynomials over finite fields." Acta Arithmetica 174.3 (2016): 239-254. <http://eudml.org/doc/286614>.

@article{WentangKuo2016,
abstract = {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 $_\{q\}[T]$, where $_\{q\}$ is a finite field of q elements. More precisely, let ω be a sequence in $_\{q\}[T]$. Given a polynomial $h ∈ _\{q\}[T]$, we define $r_\{h\}(ω) = |\{(f,g) ∈ _\{q\}[T] × _\{q\}[T]: f,g ∈ ω, f + g = h, deg f, deg g ≤ deg h, f ≠ g\}|$. We show that there exists a sequence ω of polynomials in $_\{q\}[T]$ such that $deg h ≪ r_\{h\}(ω) ≪ deg h$ for deg h tending to infinity.},
author = {Wentang Kuo, Shuntaro Yamagishi},
journal = {Acta Arithmetica},
keywords = {sidon sets; probabilistic number theory},
language = {eng},
number = {3},
pages = {239-254},
title = {On a problem of Sidon for polynomials over finite fields},
url = {http://eudml.org/doc/286614},
volume = {174},
year = {2016},
}

TY - JOUR
AU - Wentang Kuo
AU - Shuntaro Yamagishi
TI - On a problem of Sidon for polynomials over finite fields
JO - Acta Arithmetica
PY - 2016
VL - 174
IS - 3
SP - 239
EP - 254
AB - 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 $_{q}[T]$, where $_{q}$ is a finite field of q elements. More precisely, let ω be a sequence in $_{q}[T]$. Given a polynomial $h ∈ _{q}[T]$, we define $r_{h}(ω) = |{(f,g) ∈ _{q}[T] × _{q}[T]: f,g ∈ ω, f + g = h, deg f, deg g ≤ deg h, f ≠ g}|$. We show that there exists a sequence ω of polynomials in $_{q}[T]$ such that $deg h ≪ r_{h}(ω) ≪ deg h$ for deg h tending to infinity.
LA - eng
KW - sidon sets; probabilistic number theory
UR - http://eudml.org/doc/286614
ER -