Abstract
Every set A of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets \((B_i)_{i=1}^{\infty }\) such that \(B_i\) has upper Banach density 1 for all \(i \in \mathbf {N}\) and \(\sum _{i\in I} B_i \subseteq A\) for every nonempty finite set I of positive integers.
Supported in part by a grant from the PSC-CUNY Research Award Program.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Di Nasso, An elementary proof of Jin’s theorem with a bound. Electron. J. Combin. 21(2), Paper 2.37, 7 (2014)
M. Di Nasso, Embeddability properties of difference sets, Integers 14, Paper No. A 27, 24 (2014)
M. Di Nasso, I. Goldbring, R. Jin, S. Leth, M. Lupini, K. Mahlburg, On a sumset conjecture of Erdős. Canad. J. Math. 67(4), 795–809 (2015)
M.L. Gromov, Colorful categories. Uspekhi Mat. Nauk 70(4), 424, 3–76 (2015)
N. Hegyvári, On the dimension of the Hilbert cubes. J. Num. Theor. 77(2), 326–330 (1999)
N. Hegyvári, On additive and multiplicative Hilbert cubes. J. Combin. Theor. Ser. A 115(2), 354–360 (2008)
N. Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale, Proceedings of Southern Illinois Conference, Southern Illinois University, Carbondale, Ill., Lecture Notes in Mathematics, vol. 751. Springer, Berlin 1979, 119–184 (1979)
R. Jin, Standardizing nonstandard methods for upper Banach density problems, Unusual Applications of Number Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, Amer. Math. Soc., Providence, RI, 109–124 (2004)
M.B. Nathanson, Sumsets contained in infinite sets of integers. J. Combin. Theor. Ser. A 28(2), 150–155 (1980)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: Subadditivity and Limits
Appendix: Subadditivity and Limits
A real-valued arithmetic function f is subadditive if
for all \(n_1,n_2 \in \mathbf {N}\).
The following result is sometimes called Fekete’s lemma.
Lemma 2
If f is a subadditive arithmetic function, then \(\lim _{n\rightarrow \infty } f(n)/n\) exists, and
Proof
It follows by induction from inequality (4) that
for all \(n_1,\ldots , n_q \in \mathbf {N}\). Let \(f(0) = 0\). Fix a positive integer d. For all \(q,r \in \mathbf {N}_0\), we have
By the division algorithm, every nonnegative integer n can be represented uniquely in the form \(n = qd+r\), where \(q \in \mathbf {N}_0\) and \(r \in [0,d-1]\). Therefore,
Because the set \(\{f(r):r\in [0,d-1]\}\) is bounded, it follows that
for all \(d \in \mathbf {N}\), and so
This completes the proof.
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Nathanson, M.B. (2017). Sumsets Contained in Sets of Upper Banach Density 1. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-68032-3_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68030-9
Online ISBN: 978-3-319-68032-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)