Solving a ± b = 2c in elements of finite sets

Vsevolod F. Lev, and Rom Pinchasi. "Solving a ± b = 2c in elements of finite sets." Acta Arithmetica 163.2 (2014): 127-140. <http://eudml.org/doc/279495>.

@article{VsevolodF2014,
abstract = {We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates are sharp.},
author = {Vsevolod F. Lev, Rom Pinchasi},
journal = {Acta Arithmetica},
keywords = {arithmetic progression; three-term progression; sumsets},
language = {eng},
number = {2},
pages = {127-140},
title = {Solving a ± b = 2c in elements of finite sets},
url = {http://eudml.org/doc/279495},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Vsevolod F. Lev
AU - Rom Pinchasi
TI - Solving a ± b = 2c in elements of finite sets
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 2
SP - 127
EP - 140
AB - We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates are sharp.
LA - eng
KW - arithmetic progression; three-term progression; sumsets
UR - http://eudml.org/doc/279495
ER -