Abstract
In this chapter, we state and prove Jin’s Sumset Theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. We present Jin’s original nonstandard proof, as well as an ultrafilter proof due to Beiglböck and an alternative nonstandard proof due to Di Nasso. In the final section, we prove a recent quantitative strengthening of Jin’s Sumset Theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Beiglböck, An ultrafilter approach to Jin’s theorem. Isr. J. Math. 185(1), 369–374 (2011)
M. Beiglböck, V. Bergelson, A. Fish, Sumset phenomenon in countable amenable groups. Adv. Math. 223(2), 416–432 (2010)
M. Di Nasso, An elementary proof of Jin’s theorem with a bound. Electron. J. Comb. 21(2), 2.37 (2014)
M. Di Nasso, Embeddability properties of difference sets. Integers 14, A27 (2014)
M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, K. Mahlburg, High density piecewise syndeticity of sumsets. Adv. Math. 278, 1–33 (2015)
M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, K. Mahlburg, High density piecewise syndeticity of product sets in amenable groups. J. Symb. Log. 81(4), 1555–1562 (2016)
R. Jin, The sumset phenomenon. Proc. Am. Math. Soc. 130(3), 855–861 (2002)
R. Jin, Standardizing nonstandard methods for upper Banach density problems, in Unusual Applications of Number Theory. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 64 (American Mathematical Society, Providence, 2004), pp. 109–124
H.J. Keisler, S. Leth, Meager sets on the hyperfinite time line. J. Symb. Log. 56(1), 71–102 (1991)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Nasso, M.D., Goldbring, I., Lupini, M. (2019). Jin’s Sumset Theorem. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-17956-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-17955-7
Online ISBN: 978-3-030-17956-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)