101. Growth in Sumsets of Higher Convex Functions.
- Author
-
Bradshaw, Peter J.
- Subjects
CONVEX functions ,NATURAL numbers ,COMPLEX numbers ,ATOMIC displacements - Abstract
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)) . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF