A new proof of Sárközy's theorem

Author:
Neil Lyall

Journal:
Proc. Amer. Math. Soc. **141** (2013), 2253-2264

MSC (2010):
Primary 11B30

DOI:
https://doi.org/10.1090/S0002-9939-2013-11628-X

Published electronically:
March 14, 2013

MathSciNet review:
3043007

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is a striking and elegant fact (proved independently by Furstenberg and Sárközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.

**1.**A. Balog, J. Pelikán, J. Pintz, and E. Szemerédi,*Difference sets without 𝜅th powers*, Acta Math. Hungar.**65**(1994), no. 2, 165–187. MR**1278767**, https://doi.org/10.1007/BF01874311**2.**Ernie Croot and Olof Sisask,*A new proof of Roth’s theorem on arithmetic progressions*, Proc. Amer. Math. Soc.**137**(2009), no. 3, 805–809. MR**2457417**, https://doi.org/10.1090/S0002-9939-08-09594-4**3.**Harry Furstenberg,*Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions*, J. Analyse Math.**31**(1977), 204–256. MR**0498471**, https://doi.org/10.1007/BF02813304**4.**Ben Green,*On arithmetic structures in dense sets of integers*, Duke Math. J.**114**(2002), no. 2, 215–238. MR**1920188**, https://doi.org/10.1215/S0012-7094-02-11422-7**5.**W. T. GOWERS,*Additive and Combinatorial Number Theory*,`www.dpmms.cam.ac.uk/wtg10/addnoth.notes.dvi`**6.**Mariah Hamel and Izabella Łaba,*Arithmetic structures in random sets*, Integers**8**(2008), A04, 21. MR**2373088****7.**M. HAMEL, N. LYALL, AND A. RICE,*Improved bounds on Sárközy's theorem for quadratic polynomials*, to appear in Int. Math. Res. Not. 2012, doi:10.1093/imrn/rns106.**8.**T. Kamae and M. Mendès France,*van der Corput’s difference theorem*, Israel J. Math.**31**(1978), no. 3-4, 335–342. MR**516154**, https://doi.org/10.1007/BF02761498**9.**Jason Lucier,*Intersective sets given by a polynomial*, Acta Arith.**123**(2006), no. 1, 57–95. MR**2232502**, https://doi.org/10.4064/aa123-1-4**10.**N. LYALL,*A simple proof of Sárközy's theorem*,`arxiv.org/abs/1107.0243`.**11.**Neil Lyall and Ákos Magyar,*Polynomial configurations in difference sets*, J. Number Theory**129**(2009), no. 2, 439–450. MR**2473891**, https://doi.org/10.1016/j.jnt.2008.05.003**12.**N. LYALL AND ´A. MAGYAR,*Polynomial configurations in difference sets*(Revised version),`arxiv.org/abs/0903.4504`.**13.**N. LYALL AND ´A. MAGYAR,*Sárközy's Theorem*,`www.math.uga.edu/lyall/Research/Sarkozy.pdf`.**14.**Hugh L. Montgomery,*Ten lectures on the interface between analytic number theory and harmonic analysis*, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR**1297543****15.**János Pintz, W. L. Steiger, and Endre Szemerédi,*On sets of natural numbers whose difference set contains no squares*, J. London Math. Soc. (2)**37**(1988), no. 2, 219–231. MR**928519**, https://doi.org/10.1112/jlms/s2-37.2.219**16.**I. Z. Ruzsa,*Difference sets without squares*, Period. Math. Hungar.**15**(1984), no. 3, 205–209. MR**756185**, https://doi.org/10.1007/BF02454169**17.**A. Sárközy,*On difference sets of sequences of integers. III*, Acta Math. Acad. Sci. Hungar.**31**(1978), no. 3-4, 355–386. MR**487031**, https://doi.org/10.1007/BF01901984**18.**P. Varnavides,*On certain sets of positive density*, J. London Math. Soc.**34**(1959), 358–360. MR**0106865**, https://doi.org/10.1112/jlms/s1-34.3.358

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11B30

Retrieve articles in all journals with MSC (2010): 11B30

Additional Information

**Neil Lyall**

Affiliation:
Department of Mathematics, The University of Georgia, Athens, Georgia 30602

Email:
lyall@math.uga.edu

DOI:
https://doi.org/10.1090/S0002-9939-2013-11628-X

Received by editor(s):
October 18, 2011

Published electronically:
March 14, 2013

Dedicated:
Dedicated to Steve Wainger on the occasion of his retirement

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.