Solving in elements of finite sets
Tom 163 / 2014
Acta Arithmetica 163 (2014), 127-140
MSC: 11P70, 11B30.
DOI: 10.4064/aa163-2-3
Streszczenie
We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c)\in A\times B\times (A\cup B) with a+b=2c is at most (0.15+o(1))(|A|+|B|)^2 as |A|+|B|\to \infty . As a corollary, if A is antisymmetric (that is, A\cap (-A)=\emptyset ), then there are at most (0.3+o(1))|A|^2 triples (a,b,c) with a,b,c\in 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\in A and a-b=2c is at most (0.5+o(1))|A|^2. These estimates are sharp.