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###### Phrases Portable
• One can see from an elementary argument that if B is an additive basis of order h, then
• If f ( n ) is always positive for sufficiently large n, then B is called an additive basis ( of order 2 ).
• 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.
• We say that B is an additive basis of order h if r _ { B, h } ( n ) > 0 for all n sufficiently large.
• 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.
• Linnik showed that an essential component need not be an additive basis as he constructed an essential component that has " x " o ( 1 ) elements less than " x ".
• In particular, this implies that there exists an additive basis B such that r _ B ( n ) = n ^ { 1 / 2 + o ( 1 ) }, which is essentially best possible.
• A subset B is called an ( asymptotic ) additive basis of finite order if there is some positive integer h such that every sufficiently large positive integer n can be written as the sum of at most h elements of B.
• This was soon simplified and extended by ErdQs, who showed, that if " A " is any sequence with Schnirelmann density ? and " B " is an additive basis of order " k " then
• Later, ErdQs set out to answer the following question posed by Sidon : how close to the lower bound | B \ cap [ 1, n ] | \ geq n ^ { 1 / h } can an additive basis B of order h get?
• ErdQs proved that there exists an additive basis B of order 2 and constants c _ 1, c _ 2 > 0 such that c _ 1 \ log n \ leq r _ B ( n ) \ leq c _ 2 \ log n for all n sufficiently large.
• The supercharges in every super-Poincar?algebra are generated by a multiplicative basis of " m " fundamental supercharges, and an additive basis of the supercharges ( this definition of supercharges is a bit more broad than that given above ) is given by a product of any subset of these " m " fundamental supercharges.
• Indeed, they conjectured a " negative " answer to this question, namely that if B is an additive basis of order h of the natural numbers, then it cannot represent positive integers as a sum of at most h too efficiently; the number of representations of n, as a function of n, must tend to infinity.