# 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

 $r^{\prime}_{A,2}(n)=\sum_{\begin{subarray}{c}a_{1}+a_{2}=n\\ a_{1}\leq a_{2}\end{subarray}}1$

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 $r^{\prime}_{A,2}(n)=1$.

## References

• 1 Heini Halberstam and Klaus Friedrich Roth. Springer-Verlag, second edition, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001Zbl 0498.10001.
• 2 Melvyn B. Nathanson. Every function is the representation function of an additive basis for the integers. http://front.math.ucdavis.edu/math.NT/0302091arXiv:math.NT/0302091.
Title Erdős-Turan conjecture ErdHosTuranConjecture 2013-03-22 13:27:11 2013-03-22 13:27:11 bbukh (348) bbukh (348) 8 bbukh (348) Conjecture msc 11B13 msc 11B34 msc 11B05 SidonSet