Higher convexity and iterated sum sets
Abstract
Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type implications, e.g. that $A+A$ being small implies unbounded growth for a many enough times iterated product set $A \cdots A$.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2005.00125
- arXiv:
- arXiv:2005.00125
- Bibcode:
- 2020arXiv200500125H
- Keywords:
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- Mathematics - Number Theory