Lower bounds for a conjecture of Erdős and Turán (Q2839285)

Les \(A\) be a subset of \(\mathbb N:=\{0,1,\dots\}\). The \textit{representation function} of \(A\) is defined by \(r(n):=|\{(x,y)\in A^2;\,n=x+y\}|,\) where \(n\) belongs to \(\mathbb N\). Put \(A+A:=\{x+y;(x,y)\in A^2\}.\) The Erdős-Turán conjecture says that ``if \(\mathbb N\setminus (A+A)\) is finite (in other words: if \(A\) is an asymptotic basis of order 2), then the representation function \(r\) of \(A\) is unbounded (in other words: \(\limsup_{n\rightarrow +\infty} r(n)=+\infty)\).''NEWLINENEWLINEIt was proved by \textit{P. Borwein} et al. [Math. Comput. 75, No. 253, 475--484 (2006; Zbl 1093.11018)] that under the hypothesis ``\(\mathbb N\setminus (A+A)\) empty,'' one has \(\limsup_{n\rightarrow +\infty} r(n)\geq 8\).NEWLINENEWLINEUsing the generating functions analytic approach, especially as presented by \textit{D. J. Newman} in [Analytic Number Theory. New York: Springer (1998; Zbl 0887.11001)], the author proves a result of the same type (with 6 in place of 8) under a weaker hypothesis, even much more weaker than in the initial Erdős-Turán conjecture. Precisely, the author proves the following theorem.NEWLINENEWLINETheorem: Suppose that the upper asymptotic density of \(\mathbb N\setminus (A+A)\) is less than 1/10 (in other words: \({\overline d}(\mathbb N\setminus (A+A))<{1\over{10}}\), where \({\overline d}B:=\limsup_{n\rightarrow +\infty}{1\over n}|B\cap[1,n]|)\). Then NEWLINE\[NEWLINE\limsup_{n\rightarrow +\infty} r(n)\geq 6.NEWLINE\]