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Statement

Setting (p. 1). $A$ is a set of nonnegative integers and $hA$ the set of sums of exactly $h$ elements of $A$, repetitions allowed; $A$ is an asymptotic basis of order $h$ if $hA$ contains every sufficiently large integer (p. 1). The counting function $A(x)$ counts the positive elements of $A$ up to $x$ (p. 1).

Thin bases (p. 3). Every asymptotic basis of order $h$ has $A(x)\gg x^{1/h}$. An additive basis of order $h$ is called thin if $A(x)\ll x^{1/h}$. The survey says thin bases exist, the first examples being those of Raikov and of Stöhr in the 1930s, with a later class due to Cassels.

Minimal asymptotic bases (p. 3). An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$; the survey glosses this as: removing any element of $A$ destroys every representation of infinitely many integers. Nathanson constructed asymptotic bases of order 2 that are both thin and minimal. The first definition is credited to Stöhr, and Härtter gave a non-constructive proof that there are uncountably many minimal asymptotic bases of order $h$ for every $h\ge2$.

Maximal asymptotic nonbases (p. 3). $A$ is an asymptotic nonbasis of order $h$ if it is not an asymptotic basis of order $h$, that is, infinitely many positive integers lie outside $hA$. Such an $A$ is maximal if $A\cup\{b\}$ is an asymptotic basis of order $h$ for every nonnegative integer $b\notin A$. The even nonnegative integers are a maximal nonbasis of order $h$ for every $h\ge2$, and many unions of the nonnegative parts of congruence classes are others; the survey says nontrivial examples are difficult to construct. Section 4 (p. 4) records that nontrivial maximal asymptotic nonbases of every order $h\ge2$ exist (Erdős and Nathanson; Deshouillers and Grekos).

Source. Melvyn B. Nathanson, Paul Erdős and additive bases, arXiv:1401.7598v1 (2014), Section 1, p. 1, Section 2, p. 1, Section 3, p. 3, and Section 4, p. 4. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the existence statements were read clause by clause on the printed pages. Nothing here is independently reviewed.

Proof pointer

None: the survey states the existence results with references only, to Raikov, Stöhr and Cassels for thin bases, to Härtter (J. Reine Angew. Math. 214/215, 1964) for minimal bases, and to Nathanson's first paper (J. Number Theory 6, 1974) for the problems on minimal bases and maximal nonbases.

Dependencies

None.

Bears on

• Problem 326: the problem asks for a minimal basis $a_1<a_2<\cdots$ of order 2 with $a_k/k^2\to c\neq0$. A thin minimal basis of order 2, which the survey says Nathanson constructed, has $a_k\gg k^2$ by $A(x)\ll x^{1/2}$ and $a_k\ll k^2$ by $A(x)\gg x^{1/2}$ (an observation of this page); the survey says nothing on whether $a_k/k^2$ converges, so it does not answer the problem.