2024/02/28 by Jonathan Cutler, Cutler, Jonathan, Luke Pebody +3
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Combinatorial Mathematics #Computability, Logic, AI Algorithms
paper · pdf · doi:10.48550/arxiv.2402.18297
Given a set of integers A and an integer k, write A+k⋅ A for the set \a+kb:a∈ A,b∈ A\. Hanson and Petridis showed that if |A+A|≤ K|A| then |A+2⋅ A|≤ K2.95|A|. At a presentation of this result, Petridis stated that the highest known value for (log(|A+2⋅ A|/|A|))/(log(|A+A|/|A|)) (bounded above by 2.95) was (log 4)/(log 3). We show that, for all ε>0, there exist A and K with |A+A|≤ K|A| but with |A+2⋅ A|≥ K2-ε|A|. Further, we analyse a method of Ruzsa, and generalise it to give continuous analogues of the sizes of sumsets, differences and dilates. We apply this method to a construction of Hennecart, Robert and Yudin to prove that, for all ε>0, there exists a set A with |A-A|≥ |A|2-ε but with |A+A|<|A|1.7354+ε. The second author would like to thank E. Papavassilopoulos for useful discussions about how to improve the efficiency of his computer searches.