Abstract
A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B 2[g]-set is a set A of integers such that no integer has more than g essentially distinct representation-s as the sum of two elements of A. It is proved that almost all small subsets of {1, 2,…, n} are B 2[g]-sets, in the sense that if B 2 [g](k, n) denotes the number of B 2[g]-sets of cardinality k contained in the interval {1,2,…, n}, then
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Supported in part by grants from the NSA Mathematical Sciences Program and the PSC-CUNY Research Award Program. This paper was written while the author was a visitor at the (alas, now defunct) AT&T Bell Laboratories in Murray Hill, New Jersey, an excellent research institution that split into AT&T Research Labs and Lucent Bell Labs, and provided another instance of a whole being greater than the sum of its parts.
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References
A. P. Godbole, S. Janson, N. W. Locantore, Jr., and R. Rapoport, Random Sidon sequences, J. Number Theory 75 (1999), 7–22.
M. B. Nathanson, Additive number theory: Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996.
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Nathanson, M.B. (2004). On the ubiquity of Sidon sets. In: Chudnovsky, D., Chudnovsky, G., Nathanson, M. (eds) Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9060-0_16
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DOI: https://doi.org/10.1007/978-1-4419-9060-0_16
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