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Erdős–Tetali Theorem
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive basis, additive bases of every order. More specifically, it states that for every fixed integer h \geq 2, there exists a subset of the natural numbers \mathcal \subseteq \mathbb satisfying r_(n) \asymp \log n, where r_(n) denotes the number of ways that a natural number ''n'' can be expressed as the sum of ''h'' elements of ''B''. The theorem is named after Paul Erdős and Prasad V. Tetali, who published it in 1990. Motivation The original motivation for this result is attributed to a problem posed by S. Sidon in 1932 on ''economical bases''. An additive basis \mathcal\subseteq\mathbb is called ''economical'' (or sometimes ''thin'') when it is an additive basis of order ''h'' and :r_(n) \ll_ n^\varepsilon for every \varepsilon > 0. In other words, these are additive bases that use as few numbers as possible to represent a given ''n'', and yet repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Principal objects of study include the sumset of two subsets and of elements from an abelian group , :A + B = \, and the -fold sumset of , :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of from the structure of : for example, determining which elements can be represented as a sum from , where ' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that contains all even numbers greater than two, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |