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New Expander Bounds from Affine Group Energy.
- Source :
-
Discrete & Computational Geometry . Sep2021, Vol. 66 Issue 2, p552-574. 23p. - Publication Year :
- 2021
-
Abstract
- The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov (On growth rate in S L 2 (F p) , the affine group and sum-product type implications, 2018. arXiv:1812.01671). We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds in Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound from Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite A ⊂ R the following superquadratic bound holds: This improves on a bound with exponent 2 that was given in Rudnev and Shkredov. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant c such that for all finite A , B ⊂ R , [ABSTRACT FROM AUTHOR]
- Subjects :
- *BINDING energy
*EXPONENTIAL sums
*STEINER systems
Subjects
Details
- Language :
- English
- ISSN :
- 01795376
- Volume :
- 66
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Discrete & Computational Geometry
- Publication Type :
- Academic Journal
- Accession number :
- 151648967
- Full Text :
- https://doi.org/10.1007/s00454-020-00194-z