Let P_1,..., P_m ∈ Z[y] be polynomials with distinct degrees, each having zero constant term. We show that any subset A of {1,..., N} with no nontrivial progressions of the form x, x + P_1(y),..., x + P_m(y) has size |A| ≪ N/(log log N)^(cP1,...,Pm) . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers
norms.
> wtf is even the point of sending me the email than if you're gonna make it incredible difficult for me to find that particular question that i really need answered now?!
> (...) The paths that lead to new ideas tend to look unpromising. If they looked promising, other people would already have explored them.
an ode to metaphorical basement dwellers and a warning to those roaming illuminated paths
Nasreddin lost his ring in the basement of his house. He went outside and began searching. A curious passerby asked what he was doing.
"I've lost my ring" he said. "Where did you lose it?" asked the man.
"In the basement of my house" he replied. "Then why not look for it there?" said the man. "Because it's dark in my basement, and I can't see a thing in there!"
> (...) But as with any hedge, you're decreasing reward when you decrease risk.
and a tool recently mentioned here for evaluating bets
https://pastebin.com/raw/ik8YPBhy