4

let positive integer $n\ge 8$ is given, show that: there exist sets $A$ with the set of is positive integer.such $|A|=n$,and such $$|A+A|>|A-A|$$ where $$A-A=\{a-b|a\in A,b\in A\},A+A=\{a+b|a\in A,b\in A\}$$

This problem seem like this How prove this set inequality $|B|\ge 2|A|^2-1$

I search some paper, Found this problem background is :

The following are called Minkowski sum and difference of sets:

$$A-A=\{a-b|a\in A,b\in A\},A+A=\{a+b|a\in A,b\in A\}$$

The other notation is the cardinality $ |X|$ of a set $ X$

But I can't prove this, maybe $n\ge 8$ is useful

Community
  • 1
math110
  • 93,304

1 Answers1

3

You will find much discussion of this question if you type "more sums than differences" into the web. For example, here's a paper by Greg Martin and Kevin O'Bryant that should answer your question, and more.

Gerry Myerson
  • 179,216