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,
.
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 , where is a finite field of q elements....
Let denote the polynomial ring over , the finite field of elements. Suppose the characteristic of is not or . We prove that there exist infinitely many such that the set contains a Sidon set which is an additive basis of order .
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define 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 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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