Let $\mathbb{F}$ be a Galois field with $p$ elements for a prime number $p$, $S_0$ be the set of squares of $\mathbb{F}$, $a$ be a non-null element of $\mathbb{F}$ and $S_1 = \{ x \in S_0 : x + a \in S_0 \}$.
Rabers proves $|S_1| = \frac{|S_0|}{2} + \epsilon$ with $|\epsilon| \leq 1$.
Now, consider $b$ a non-null element of $\mathbb{F}$ and $S_2 = \{ x \in S_1 : x + b \in S_1 \}$.
Is the property true? $|S_2| = \frac{|S_1|}{2} + \epsilon$ with $|\epsilon| \leq 1$.
More generally, can one prove that $|S_{n+1}| = \frac{|S_n|}{2} + \epsilon$?
ps: I am not very familiar with field theory.
No, it's a rather special property.
For a small counterexample, let $p=5$, $\,a=1\,$ and $\,b=2$.
Then $S_0=\{-1,0,1\}$, $\ S_1=\{-1,0\}$, but $\,S_2=\emptyset$.