The Erdos-Turan conjecture on additive bases

Importance: Outstanding ✭✭✭✭
Recomm. for undergrads: no
Prize: $500 (Erdos - Graham [EG])
Posted by: mdevos
on: June 8th, 2007

Let $B \subseteq {\mathbb N}$. The \emph{representation function} $r_B : {\mathbb N} \rightarrow {\mathbb N}$ for $B$ is given by the rule $r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \}$. We call $B$ an \emph{additive basis} if $r_B$ is never $0$.

\begin{conjecture} If $B$ is an additive basis, then $r_B$ is unbounded. \end{conjecture}

This famous conjecture seems intuitively likely, but to date, there has been relatively little progress on it despite considerable attention. Two positive results are a theorem of Dirac [D] which shows that $r_B$ cannot be constant from some point on, and a theorem of Borwein, Choi, and Chu [BCC] which shows that $r_B$ cannot be bounded above by $6$.

On the other hand, if we consider the related problem for subsets of integers instead of natural numbers, Nathanson [N] has shown that the conjecture does not hold.

Bibliography

[BCC] P. Borwein, S. Choi, and F. Chu, \href[An old conjecture of Erdos-Turan on additive bases]{http://www.ams.org/mcom/2006-75-253/S0025-5718-05-01777-1/S0025-5718-05-01777-1.pdf}, Mathematics of Computation. Volume 75, Number 253, Pages 475–484.

[D] G. A. Dirac, Note on a problem in additive number theory, J. London Math. Soc. 26 (1951), 312–313.

[EG] P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979), no. 3-4, 325–344 (1980). \MRhref{0570317}

*[ET] P. Erdos and P. Turan, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215. \MRhref{0006197}

[N] Melvyn B. Nathanson, \href[Unique representation bases for the integers]{http://front.math.ucdavis.edu/math.NT/0302091}, Acta Arith. 108 (2003), no. 1, 1–8. \MRhref{1971077}


* indicates original appearance(s) of problem.