On the number of sums and products

h-Fold Sums from a Set with Few Products

“…Further improvements to this result were achieved by Nathanson [17], Ford [12], Elekes [8], and finally Solymosi [19] and [20].…”

Untitled

“…The version over finite fields is harder, since there is no natural topology or fully natural ordering to work with. (Over R, there is a brief and very natural proof [Ele97] based on a result that is essentially topological [ST83]; the best-known bound for the sum-product theorem over R has a direct proof, also topological [Sol09]. )…”

Depth estimation via sampling

“…The insight that this lemma has something to do with incidences and similar problems is due to Székely [Szé97]. Elekes [Ele97] used the crossing lemma to prove surprising lower bounds on sum and products problems.…”