On the General Erdős-Turán Conjecture

The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfiesa n ≤ c n 2 , for some constant c > 0 and for all n , then the number of representations functions of A is unbounded. Here, we introduce the function ψ ( n ) , giving the minimum of the maximal number of representations of a finite sequence A = { a k : 1 ≤ k ≤ n } of n natural numbers satisfyinga k ≤ k 2for all k . We show that ψ ( n ) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to l i m n → ∞ ψ ( n ) = ∞ . We also compute some values of ψ ( n ) . We further introduce and study the notion of capacity, which is related to the ψ function by the fact that l i m n → ∞ ψ ( n ) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.