Erdős-Turan conjecture

Erdős-Turan conjecture asserts there exist no asymptotic basis (http://planetmath.org/AsymptoticBasis) $A\subset {\mathbb{N}}_0$ of order $2$ such that its representation function

$$r_{A,2}^{\prime }(n)=\sum _{\begin{array}{c}{a}_1+a_2=n\\ a_1\le a_2\end{array}}1$$

is bounded.

Alternatively, the question can be phrased as whether there exists a power series $F$ with coefficients $0$ and $1$ such that all coefficients of $F^2$ are greater than $0$, but are bounded.

If we replace set of nonnegative integers by the set of all integers, then the question was settled by Nathanson[2] in negative, that is, there exists a set $A\subset \mathbb{Z}$ such that $r_{A,2}^{\prime }(n)=1$.

Title	Erdős-Turan conjecture
Canonical name	ErdHosTuranConjecture
Date of creation	2013-03-22 13:27:11
Last modified on	2013-03-22 13:27:11
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	8
Author	bbukh (348)
Entry type	Conjecture
Classification	msc 11B13
Classification	msc 11B34
Classification	msc 11B05
Related topic	SidonSet