TY - JOUR
T1 - An improved construction of progression-free sets
AU - Elkin, Michael
N1 - Funding Information: ∗ Preliminary version of this paper was published in [14]. ∗∗ This research has been supported by the Israeli Academy of Science, grant 483/06. Received April 28, 2009 and in revised form August 24, 2009
PY - 2011/8/1
Y1 - 2011/8/1
N2 - The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdo{double acute}s and Turán in 1936. They have presented a construction with {pipe}S{pipe} = Ω (nlog32) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is. Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of Θ (√log n), and shows that. In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.
AB - The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdo{double acute}s and Turán in 1936. They have presented a construction with {pipe}S{pipe} = Ω (nlog32) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is. Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of Θ (√log n), and shows that. In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.
UR - http://www.scopus.com/inward/record.url?scp=79960987955&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s11856-011-0061-1
DO - https://doi.org/10.1007/s11856-011-0061-1
M3 - Article
SN - 0021-2172
VL - 184
SP - 93
EP - 128
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -