Sums, Differences and Dilates

Given a set of integers A and an integer k, write A+k· A for the set +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.