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Convexity, superquadratic growth, and dot products
Brandon Hanson1, Oliver Roche-Newton2, Steven Senger3
1University of Maine Orono Maine USA.
Journal of the London Mathematical Society
|March 22, 2024
Summary
This study improves bounds on dot products for point sets, introducing a new superquadratic expander. This mathematical advancement offers better than quadratic growth in set expansions.
Area of Science:
- Combinatorics
- Additive Combinatorics
- Harmonic Analysis
Background:
- The study of dot products and their bounds is fundamental in various mathematical fields.
- Previous research has explored set expansion properties, but achieving superquadratic growth remains a challenge.
Purpose of the Study:
- To establish an improved bound for the number of dot products determined by a finite point set.
- To introduce and analyze a novel superquadratic expander involving products and shifts.
- To demonstrate the utility of this expander in proving other set expansion results.
Main Methods:
- Developing a new superquadratic expander using products and shifts.
- Leveraging arguments from existing literature (Hanson, Roche-Newton, Rudnev).
- Employing predominantly elementary mathematical methods.
Main Results:
- An improved bound for the number of dot products determined by a point set:
. - Proof of a superquadratic expander for finite sets:
. - Demonstration of a general result concerning the growth of sets defined via convexity and sum sets.
Conclusions:
- The established improved bound offers a significant advancement in understanding dot product determination.
- The newly developed superquadratic expander has broad applicability in proving other set expansion results.
- The study highlights the power of elementary methods in addressing complex problems in additive combinatorics.
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