Growth in Sumsets of Higher Convex Functions

The main results of this paper concern growth in sums of a k -convex function f . Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71–85, 2020)) of a growth result for f ( A ) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for aligned |2ᵏ f(A) - (2ᵏ-1)f(A)|. aligned | 2 k f ( A ) - ( 2 k - 1 ) f ( A ) | . We also generalise a recent result from Hanson et al. (J Lond Math Soc, 2021), proving that for any finite A⊂ R A ⊂ R aligned | 2ᵏ f(sA-sA) - (2ᵏ-1) f(sA-sA)| ₛ |A|²ˢ aligned | 2 k f ( s A - s A ) - ( 2 k - 1 ) f ( s A - s A ) | ≫ s | A | 2 s where s = k+1/2 s = k + 1 2 . This allows us to prove that, given any natural number s ∈ N s ∈ N , there exists $$m = m(s)$$ m = m ( s ) such that if A ⊂ R A ⊂ R , then aligned |(sA-sA)⁽ᵐ⁾| ₛ |A|ˢ. aligned | ( s A - s A ) ( m ) | ≫ s