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On Sidon sets and asymptotic bases
Author(s) -
Cilleruelo Javier
Publication year - 2015
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdv050
Subject(s) - mathematics , conjecture , sequence (biology) , basis (linear algebra) , order (exchange) , integer (computer science) , combinatorics , asymptotic formula , integer sequence , discrete mathematics , generating function , geometry , genetics , finance , computer science , economics , biology , programming language
Erdös conjectured the existence of an infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We progress towards this conjecture in several directions. We prove the conjecture for all cyclic groups Z N with N large enough. We also show that there is an infiniteB 2 [ 2 ]sequence which is an asymptotic basis of order 3. Finally, we prove that, for any ε > 0 , there is a Sidon sequence which is an asymptotic basis of order 3 + ε ; that is, any positive sufficiently large integer n can be written as a sum of four elements of the sequence, one of them smaller than n ε .