Erdős–Turán conjecture on additive bases

The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941.

The question concerns subsets of the natural numbers, typically denoted by ${\displaystyle \mathbb {N} }$, called additive bases. A subset ${\displaystyle B}$ is called an (asymptotic) additive basis of finite order if there is some positive integer ${\displaystyle h}$ such that every sufficiently large positive integer ${\displaystyle n}$ can be written as the sum of at most ${\displaystyle h}$ elements of ${\displaystyle B}$. For example, the natural numbers are themselves an additive basis of order 1, since every natural number is trivially a sum of at most one natural number. It is a non-trivial theorem of Lagrange (Lagrange's four-square theorem) that the set of positive square numbers is an additive basis of order 4. Another highly non-trivial and celebrated result along these lines is Vinogradov's theorem.

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