Georgia Institute of Technology College of Sciences

Fix an integer $ k\geq 2 $ and a sufficiently large prime $ p $ depending on $ k $. We prove that any subset of $ \mathbb{F}_p^n $ with density at least $ C/(\log^{k^{O(k)}} n)^c $, where $ C,c>0 $ depend only on $ k $ and $ p $, contains a nontrivial restricted polynomial progression of the form $ x, x+y, x+y^2,\ldots,x+y^{k-1} $, where powers are taken coordinatewise and the coordinates of the nonzero vector $ y $ belong to the fixed set $ \{0,2,3,\ldots,M\} $, with $ M $ depending only on $ k $. 

The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.