Remark: Family 182. Power savings for polynomial-difference-free sets

For every fixed intersective integer polynomial $h$ of degree $k\ge 2$ with positive leading coefficient, proves that a subset of $\{1,\dots ,N\}$ avoiding nonzero values $h(1),h(2),\dots$ as differences has size $O_h(N^{1-c_k})$, with $c_k>0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when $h$ has a unit root modulo every modulus, with exponent allowed to depend on $h$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.