A Basic Question About Multiplicative Energy

As part of some research project, I got to a basic question about multiplicative energy. Embarrassingly , I wasn’t able to get any non-trivial bound for it. Here is the problem. Any information about it would be highly appreciated.

Problem. Let $A=\{1,2,3,\ldots,n^{1/2}\}$. Let $B$ be a set of $n$ real numbers. How large can the multiplicative energy $E^\times(A,B)$ be?

• Can we prove that $E^\times(A,B)$ must be asymptotically smaller than $n^{2}$ ?
• Is there a $B$ for which $E^\times(A,B)$ is asymptotically larger than $n^{3/2}$?

Both questions are about improving the exponent of $n$. Sub-polynomial improvements would be less interesting. Multiplicative energy means

$E^\times(A,B) = |\{(a,a',b,b')\in A^2\times B^2\ :\ ab=a'b'\}|$

Any ideas?

A variant. Still here and want to read more? We can also consider the case when $A$ is larger. Let $A=\{1,2,3,\ldots,n^{\alpha}\}$, where $1/2 < \alpha <1$.

In this case, a simple argument leads to an upper bound that is better than the trivial one. Let $r_A(x) = |\{(a,a')\in A^2:\ a/a'=x \}|$. The Cauchy-Schwarz inequality implies that

$E^\times(A,B) = |\{(a,a',b,b')\in A^2\times B^2\ :\ a/a'=b'/b\}|$

$= \sum_{x\in A/A} r_A(x) \cdot r_B(x) \le \sqrt{\sum_{x}r_A(x)^2} \cdot \sqrt{\sum_x r_B(x)^2}$

$=\sqrt{E^\times(A) \cdot E^\times(B)}$

It is not difficult to show that $E^\times(A)=\Theta(n^{2\alpha})$, up to sub-polynomial factors. This implies that $E^\times(A,B)=O(n^{3/2+\alpha})$, up to sub-polynomial factors. In this case, the trivial upper bound is $E^\times(A,B)=O(n^{1+2\alpha})$.

The above still leaves a large gap between the lower bound $O(n^{1+\alpha})$ and the upper bound $E^\times(A,B)=O(n^{3/2+\alpha})$, up to sub-polynomial factors.

Any ideas?

2 thoughts on “A Basic Question About Multiplicative Energy”

1. Ilya Mironov |

Why wouldn’t a random set B (say, n numbers sampled from [0,1]) match n^2 almost surely? The collision ab=a’b’ implies that a/a’ = b’/b, but a/a’ only take fewer than n values are b’/b are unrestricted. Why would we expect any collisions between the A/A and B/B sets?

• Right. With a random set we expect the multiplicative energy to be minimal. Since |A|=n^{1/2} and |B|=n, a minimal multiplicative energy has size of about n^{3/2}. In this case, n^2 is the maximum size.