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Statement

Setting (pp. 1--3). For a set $A$ of nonnegative integers, $A$ is an asymptotic basis of order $h$ if the sumset $hA$ of sums of exactly $h$ elements of $A$, repetitions allowed, contains every sufficiently large integer (p. 1). In the passage the survey quotes from Erdős and Turán (1941), $f(n)$ is the number of representations of $n$ as $a_i+a_j$ and $g(n)$ the number of representations of $n$ as $a_ia_j$ (p. 3).

Conjecture (p. 3, quoted). "The Erdős-Turán conjecture, that the representation function of an asymptotic basis of order 2 is always unbounded, is a major unsolved problem in additive number theory."

In the quoted closing passage of Erdős and Turán (p. 3) the conjecture reads: if $f(n)>0$ for $n>n_0$, then $\limsup f(n)=\infty$. The same passage says the corresponding result for $g(n)$ can be proved. The survey records that Erdős published that multiplicative proof in 1964, that Nešetřil and Rödl simplified it, and that Nathanson generalized it (p. 3).

Status in the survey. Open as of the survey's date (January 2014). The survey adds (p. 3) that Nathanson, looking for a counterexample, built asymptotic bases of order 2 that are both thin and minimal, none of which is a counterexample; the definitions are on [[additive_bases/nathanson_2014_paul_erdos_additive_bases/definition_p3|the definitions page]].

Source. Melvyn B. Nathanson, Paul Erdős and additive bases, arXiv:1401.7598v1 (2014), Section 1, p. 1, and Section 3, pp. 2--3. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the quoted passage were read clause by clause on the printed page. Nothing here is independently reviewed.

Proof pointer

None: an open conjecture. For the multiplicative analogue the survey points to Erdős, On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261.

Dependencies

None.

Bears on

• Problem 28: the problem is the Erdős-Turán conjecture as the survey states it, with $1_A\ast1_A(n)$ for the representation function; whether ordered or unordered pairs are counted does not change whether it is bounded. The survey records the problem as open in 2014 and gives no progress on it.