Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields

We prove new lower bounds on the maximum size of subsets A⊆ \1,,N\ or A⊆ Fₚⁿ not containing three-term arithmetic progressions. In the setting of \1,,N\, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of Fₚⁿ for a fixed prime p and large n, we prove a lower bound of (cp)ⁿ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem).