Given a set A ⊂ ℕ let denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and for all n̅ ∈ ℤₘ.
Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and for all n̅ ∈ Zₘ.
For any positive integer k and any set A of nonnegative integers, let denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both and hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying for all n ≥ n₀, we have as n → ∞.