3 Stabilizer
\(\operatorname{\operatorname {Stab}}_K(A)\) is the set \(\{ x | |A + x A| \leq K |A|\} \).
For \(a \in \operatorname{\operatorname {Stab}}_K(A)\), we also have \(a^{-1} \in \operatorname{\operatorname {Stab}}_K(A)\).
If \(a = 0\) this is trivial, otherwise by 1.2 we have \(|A + a^{-1}A| = |a(A + a^{-1}A)| = |A + a A| \leq K |A|\).
If \(A \neq \emptyset \) and \(a \in \operatorname{\operatorname {Stab}}_K(A)\) then \(1 \leq K\).
Trivial.
For \(a \in \operatorname{\operatorname {Stab}}_K(A)\), we also have \(a \in \operatorname{\operatorname {Stab}}_{K^3}(A)\).
We have \(-\operatorname{\operatorname {Stab}}_{K}(A) \subseteq \operatorname{\operatorname {Stab}}_{K^3}(A)\).
Immediate from 3.4.
For \(a \in \operatorname{\operatorname {Stab}}_{K_1}(A), b \in \operatorname{\operatorname {Stab}}_{K_2}(A)\), we have \(a+b \in \operatorname{\operatorname {Stab}}_{K_1^8 K_2}(A)\).
We have \(\operatorname{\operatorname {Stab}}_{K_1}(A) + \operatorname{\operatorname {Stab}}_{K_2}(A) \subseteq \operatorname{\operatorname {Stab}}_{K_1^8 K_2}(A)\).
Immediate from 3.6.
For \(n \in \mathbb {N}\) we have \((n+1) \cdot \operatorname{\operatorname {Stab}}_{K}(A) \subseteq \operatorname{\operatorname {Stab}}_{K^{8n + 1}}(A)\).
By induction with 3.7.
We have \(\operatorname{\operatorname {Stab}}_{K_1}(A) - \operatorname{\operatorname {Stab}}_{K_2}(A) \subseteq \operatorname{\operatorname {Stab}}_{K_1^8 K_2^3}(A)\).
For \(a \in \operatorname{\operatorname {Stab}}_{K_1}(A), b \in \operatorname{\operatorname {Stab}}_{K_2}(A)\), we have \(ab \in \operatorname{\operatorname {Stab}}_{K_1 K_2}(A)\).
We have \(\operatorname{\operatorname {Stab}}_{K_1}(A) \operatorname{\operatorname {Stab}}_{K_2}(A) \subseteq \operatorname{\operatorname {Stab}}_{K_1 K_2}(A)\).
Immediate from 3.10.
If \(a \in \operatorname{\operatorname {Stab}}_K(A)\) and \(K \leq K'\) then \(a \in \operatorname{\operatorname {Stab}}_{K'}(A)\).
Trivial from the definition.
If \(K \leq K'\) then \(\operatorname{\operatorname {Stab}}_K(A) \subseteq \operatorname{\operatorname {Stab}}_{K'}(A)\).
Trivial from 3.12
If \(1 \leq K\) implies \(K \leq K'\) then \(\operatorname{\operatorname {Stab}}_K(A) \subseteq \operatorname{\operatorname {Stab}}_{K'}(A)\).
We have \(3 \operatorname{\operatorname {Stab}}_K(A)^2 - 3 \operatorname{\operatorname {Stab}}_K(A)^2 \subseteq \operatorname{\operatorname {Stab}}_{K^{374}}(A)\).
We have \(\frac{\min (|\operatorname{\operatorname {Stab}}_K(A)|^2, p)}2 \leq |\operatorname{\operatorname {Stab}}_{K^{374}}(A)|\).
If \(4 \leq |\operatorname{\operatorname {Stab}}_K(A)|\), then \(\min (|\operatorname{\operatorname {Stab}}_K(A)|^{\frac32}, \frac p2) \leq |\operatorname{\operatorname {Stab}}_{K^{374}}(A)|\).
From direct calculation using 3.16.
If \(4 \leq |\operatorname{\operatorname {Stab}}_K(A)|\) for all \(n \in \mathbb {N}\), \(\min (|\operatorname{\operatorname {Stab}}_K(A)|^{\left(\frac32\right)^n}, \frac p2) \leq |\operatorname{\operatorname {Stab}}_{K^{374^n}}(A)|\).
By induction on 3.17.
\(\mathrm{StabC}_2(\beta ) = 374^{\lceil \log _{\frac32}(1 / \beta ) \rceil }\)
If \(4 \leq |\operatorname{\operatorname {Stab}}_K(A)|\) and \(p^\beta \leq |\operatorname{\operatorname {Stab}}_K(A)|\), then \(\frac{p}2 \leq |\operatorname{\operatorname {Stab}}_{K^{\mathrm{StabC}_2(\beta )}}(A)|\).
By setting \(n = \mathrm{StabC}_2(\beta )\) at 3.18.
\(\mathrm{StabC}(\beta ) = 9 \mathrm{StabC}_2(\beta )\)
If \(4 \leq |\operatorname{\operatorname {Stab}}_K(A)|\) and \(p^\beta \leq |\operatorname{\operatorname {Stab}}_K(A)|\), then \(\operatorname{\operatorname {Stab}}_{K^{\mathrm{StabC}(\beta )}} (A) = \mathbb {F}\).
By Cauchy-Davenport and 3.7 after ??.
If \(p^\beta \leq |A| \leq p^{1 - \beta }\) and \(K {\lt} \frac{p^\beta }2\), then \(\operatorname{\operatorname {Stab}}_K(A) \neq \mathbb {F}\).
2.1 gives a value \(a\) which by direct computation we can show isn’t in \(\operatorname{\operatorname {Stab}}_K(A)\).
If \(4 \leq |\operatorname{\operatorname {Stab}}_K(A)|, p^\beta \leq |\operatorname{\operatorname {Stab}}_K(A)|, p^\gamma \leq |A| \leq p^{1 - \gamma }\) then \(\frac{p^\gamma }2 \leq K^{\mathrm{StabC}(\beta )}\)