# On the structure of sets with small doubling property on the plane (I)

Acta Arithmetica (1998)

- Volume: 83, Issue: 2, page 127-141
- ISSN: 0065-1036

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topYonutz Stanchescu. "On the structure of sets with small doubling property on the plane (I)." Acta Arithmetica 83.2 (1998): 127-141. <http://eudml.org/doc/207110>.

@article{YonutzStanchescu1998,

abstract = {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.},

author = {Yonutz Stanchescu},

journal = {Acta Arithmetica},

keywords = {additive number theory; small doubling property; sumset; lattice points},

language = {eng},

number = {2},

pages = {127-141},

title = {On the structure of sets with small doubling property on the plane (I)},

url = {http://eudml.org/doc/207110},

volume = {83},

year = {1998},

}

TY - JOUR

AU - Yonutz Stanchescu

TI - On the structure of sets with small doubling property on the plane (I)

JO - Acta Arithmetica

PY - 1998

VL - 83

IS - 2

SP - 127

EP - 141

AB - 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.

LA - eng

KW - additive number theory; small doubling property; sumset; lattice points

UR - http://eudml.org/doc/207110

ER -

## References

top- [B] Y. Bilu, Structure of sets with small sumsets, Mathématiques Stochastiques, Univ. Bordeaux 2, Preprint 94-10, Bordeaux, 1994.
- [F1] G. A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monographs 37, Amer. Math. Soc., Providence, R.I., 1973.
- [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).
- [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.
- [L-S] V. F. Lev and P. Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1995), 85-91. Zbl0817.11005
- [R] I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379-388. Zbl0816.11008
- [S] Y. Stanchescu, On addition of two distinct sets of integers, Acta Arith. 75 (1996), 191-194.

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