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?

Isn’t this supposed to be an easy problem?!

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. 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.

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