Representation Functions With Prescribed Rates of Growth
Abstract
ABSTRACT Fix an integer , and let be (not necessarily distinct) positive integers with . For any subset , let denote the number of solutions to the equation Given a function satisfying , we ask: when does there exist a set such that ? We prove that this is always possible when is regularly varying and satisfies . If one only requires , much weaker regularity conditions suffice: we show that such a set exists for every increasing function satisfying and . Finally, we give a probabilistic heuristic supporting the following: if satisfies , then for infinitely many .
Fields of Study & Subjects
Limits and Structures in Graph Theory
Advanced Topology and Set Theory
Economic theories and models
Publication details
- Journal / source
- Random Structures and Algorithms
- Volume
- 69
- Issue
- 1
- Publication date
- 1 August 2026
- Language
- EN