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 . Let be a set of real numbers. How large can the multiplicative energy be?
- Can we prove that must be asymptotically smaller than ?
- Is there a for which is asymptotically larger than ?
Both questions are about improving the exponent of . Sub-polynomial improvements would be less interesting. Multiplicative energy means
A variant. Still here and want to read more? We can also consider the case when is larger. Let , where .
In this case, a simple argument leads to an upper bound that is better than the trivial one. Let . The Cauchy-Schwarz inequality implies that
It is not difficult to show that , up to sub-polynomial factors. This implies that , up to sub-polynomial factors. In this case, the trivial upper bound is .
The above still leaves a large gap between the lower bound and the upper bound , up to sub-polynomial factors.