Additive basis

In additive number theory, an additive basis is a set ${\displaystyle S}$ of natural numbers with the property that, for some finite number ${\displaystyle k}$, every natural number can be expressed as a sum of ${\displaystyle k}$ or fewer elements of ${\displaystyle S}$. That is, the sumset of ${\displaystyle k}$ copies of ${\displaystyle S}$ consists of all natural numbers. The order or degree of an additive basis is the number ${\displaystyle k}$. When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set ${\displaystyle S}$ for which all but finitely many natural numbers can be expressed as a sum of ${\displaystyle k}$ or fewer elements of ${\displaystyle S}$.^[1]

For example, by Lagrange's four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal numbers for ${\displaystyle k}$-sided polygons form an additive basis of order ${\displaystyle k}$. Similarly, the solutions to Waring's problem imply that the ${\displaystyle k}$th powers are an additive basis, although their order is more than ${\displaystyle k}$. By Vinogradov's theorem, the prime numbers are an asymptotic additive basis of order at most four, and Goldbach's conjecture would imply that their order is three.^[1]

The unproven Erdős–Turán conjecture on additive bases states that, for any additive basis of order ${\displaystyle k}$, the number of representations of the number ${\displaystyle n}$ as a sum of ${\displaystyle k}$ elements of the basis tends to infinity in the limit as ${\displaystyle n}$ goes to infinity. (More precisely, the number of representations has no finite supremum.)^[2] The related Erdős–Fuchs theorem states that the number of representations cannot be close to a linear function.^[3] The Erdős–Tetali theorem states that, for every ${\displaystyle k}$, there exists an additive basis of order ${\displaystyle k}$ whose number of representations of each ${\displaystyle n}$ is ${\displaystyle \Theta (\log n)}$.^[4]

A theorem of Lev Schnirelmann states that any sequence with positive Schnirelmann density is an additive basis. This follows from a stronger theorem of Henry Mann according to which the Schnirelmann density of a sum of two sequences is at least the sum of their Schnirelmann densities, unless their sum consists of all natural numbers. Thus, any sequence of Schnirelmann density ${\displaystyle \varepsilon >0}$ is an additive basis of order at most ${\displaystyle \lceil 1/\varepsilon \rceil }$.^[5]

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