A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets)

Theorem 1.1: absolute $c>0$, $C$ with $|A|\le CN^{1-c}$ for every square-difference-free $A\subseteq[N]$; consequently the square-difference graph on $[N]$ has chromatic number at least $C^{-1}N^c$. Companions: for an intersective $h$ of degree $k\ge2$, a power saving $O_h(N^{1-c_k})$ with exponent depending only on $k$; and for $h$ with a unit root modulo every modulus, a power saving $C_hN^{1-c_h}$ for differences $h(p)$ at primes. The exponent is tiny and not computed, far from Ruzsa's lower-bound exponent $0.733$; the true exponent is not addressed.

Theorem 1.1: absolute $c>0$, $C$ with $|A|\le CN^{1-c}$ for every square-difference-free $A\subseteq[N]$; consequently the square-difference graph on $[N]$ has chromatic number at least $C^{-1}N^c$. Companions: for an intersective $h$ of degree $k\ge2$, a power saving $O_h(N^{1-c_k})$ with exponent depending only on $k$; and for $h$ with a unit root modulo every modulus, a power saving $C_hN^{1-c_h}$ for differences $h(p)$ at primes. The exponent is tiny and not computed, far from Ruzsa's lower-bound exponent $0.733$; the true exponent is not addressed.