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 ${r}_{A,2}^{\prime}(n)=1$.
References
- 1 Heini Halberstam and Klaus Friedrich Roth. Sequences. 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 |
---|---|
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 |