Abstract
Szemeredi’s theorem says that relatively dense sets contain arithmetic progressions. The purpose of this chapter is to present a result of Leth from Leth (Proc Am Math Soc 134:1579–1589, 2006) which shows that certain sparse sets contain “near” arithmetic progressions. We then detail the connection between the aforementioned theorem of Leth and the Erdős-Turán conjecture.
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Notes
- 1.
In this chapter, we deviate somewhat from our conventions so as to match up with the notation from [89].
References
P. Erdős, P. Turán, On some sequences of integers. J. Lond. Math. Soc. S1-11(4), 261 (1936)
P. Erdös, P. Turán, On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16, 212–215 (1941)
B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167(2), 481–547 (2008)
S. Leth, Near arithmetic progressions in sparse sets. Proc. Am. Math. Soc. 134(6), 1579–1589 (2006)
S. Leth, Nonstandard methods and the erdös-turán conjecture, in The Strength of Nonstandard Analysis (Springer, Berlin, 2007), pp. 133–142
S. Vijay, On the discrepancy of quasi-progressions. Electron. J. Comb. 15(1), Research Paper 104, 14 (2008)
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Near Arithmetic Progressions in Sparse Sets. 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_14
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DOI: https://doi.org/10.1007/978-3-030-17956-4_14
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