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,
$li{m}_{n\to \infty}r\u2099\left(\omega \right)/{n}^{\u03f5}=0$.
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 ${}_{q}\left[T\right]$, where ${}_{q}$ is a finite field of q elements....

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

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define ${r}_{N}^{\left(m\right)}\left(\omega \right)$ 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 ${r}_{N}^{\left(m\right)}\left(\omega \right)<K$ 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.

Download Results (CSV)