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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.

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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 $\,S_2=\emptyset$.

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