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).