تقرير
On an entropic analogue of additive energy
| العنوان: | On an entropic analogue of additive energy |
|---|---|
| المؤلفون: | Goh, Marcel K. |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Number Theory, 11B13, 94A17 |
| الوصف: | Recent advances have linked various statements involving sumsets and cardinalities with corresponding statements involving sums of random variables and entropies. In this vein, this paper shows that the quantity $2{\bf H}\{X, Y\} - {\bf H}\{X+Y\}$ is a natural entropic analogue of the additive energy $E(A,B)$ between two sets. We develop some basic theory surrounding this quantity, and demonstrate its role in the proof of Tao's entropy variant of the Balog--Szemer\'edi--Gowers theorem. We examine the regime where entropic additive energy is small, and discuss a family of random variables related to Sidon sets. In finite fields, one can define an entropic multiplicative energy as well, and we formulate sum-product-type conjectures relating these two entropic energies. Comment: 19 pages, Conjecture 15 in previous version is a quick consequence of a recent theorem of M\'ath\'e and O'Regan |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.18798 |
| رقم الأكسشن: | edsarx.2406.18798 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |