Let K be a finite set of lattice points in a plane. We prove that if |K| is sufficiently large and |K+K| < (4 - 2/s)|K| - (2s-1), then there exist s - 1 parallel lines which cover K. We also obtain some more precise structure theorems for the cases s = 3 and s = 4.
@article{bwmeta1.element.bwnjournal-article-aav83i2p127bwm,
author = {Yonutz Stanchescu},
title = {On the structure of sets with small doubling property on the plane (I)},
journal = {Acta Arithmetica},
volume = {84},
year = {1998},
pages = {127-141},
zbl = {0885.11018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav83i2p127bwm}
}
Yonutz Stanchescu. On the structure of sets with small doubling property on the plane (I). Acta Arithmetica, Tome 84 (1998) pp. 127-141. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav83i2p127bwm/
[000] [B] Y. Bilu, Structure of sets with small sumsets, Mathématiques Stochastiques, Univ. Bordeaux 2, Preprint 94-10, Bordeaux, 1994.
[001] [F1] G. A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monographs 37, Amer. Math. Soc., Providence, R.I., 1973.
[002] [F2] G. A. Freiman, Inverse problems in additive number theory VI. On the addition of finite sets III, Izv. Vyssh. Uchebn. Zaved. Mat. 1962, no. 3 (28), 151-157 (in Russian).
[003] [F3] G. A. Freiman, What is the structure of K if K + K is small?, in: Lecture Notes in Math. 1240, Springer, 1987, New York, 109-134.
[004] [L-S] V. F. Lev and P. Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1995), 85-91. | Zbl 0817.11005
[005] [R] I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379-388. | Zbl 0816.11008
[006] [S] Y. Stanchescu, On addition of two distinct sets of integers, Acta Arith. 75 (1996), 191-194.