Growth in sumsets of higher convex functions (Q6081398)

In the first part, the author obtains optimal bounds which improve on \textit{B. Hanson} et al. [Combinatorica 42, No. 1, 71--85 (2022; Zbl 1513.11024)] by logarithmic factors. In particular, it is proved that, for any \(k\)-convex function \(f\) and any finite set \(A\) of real numbers such that \(\vert A+A-A\vert \le K\vert A\vert \) for some constant \(K\), we have \[\vert 2^k f(A)-(2^k-1)f(A)\vert \gg \frac{\vert A\vert ^{2^{k+1}-1}}{\vert A+A-A\vert ^{2^{k+1}-k-2}}\ge \vert A\vert ^{k+1} K^{-(2^{k+1}-k-2}.\] In particular, a simple proof is given of the inequality \[\vert A+A-A\vert \vert f(A)+f(A)-f(A)\vert \gg \vert A\vert ^3.\] In the second part, the author proves the two lower bounds \(\vert A+A-A\vert \vert AA\vert ^2\gg \vert A\vert ^4\) when \(A\) is a finite set of complex numbers and \(\vert A+A-A\vert ^3\vert AA\vert ^4\gg \varepsilon \vert A\vert ^{9-\varepsilon}/q^2\) for any finite subset \(A\) of \(\mathbb{F}_q((1/t))\).