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 | 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 | 2 k f ( s A - s A ) - ( 2 k - 1 ) f ( s A - s A ) | ≫ s | A | 2 s where s = k + 1 2 . This allows us to prove that, given any natural number s ∈ N , there exists m = m ( s ) such that if A ⊂ R , then 1 | ( s A - s A ) ( m ) | ≫ s | A | s . This is progress towards a conjecture (Balog et al. in Electron J Comb 24(3):Paper No. 3.14, 17, 2017) which states that (1) can be replaced with | ( A - A ) ( m ) | ≫ s | A | s . Developing methods of Solymosi, and Bloom and Jones, and using an idea from Bradshaw et al. (Electron J Comb 29, 2021), we present some new sum-product type results in the complex numbers C and in the function field F q ( ( t - 1 ) ) .