We’ll see three propositions as the opening remarks of this topic. As the first one, we’re checking:
Proposition 1.
- If \({\mathcal F} \) satisfies the \( \Gamma(b) \)-condition on the norm \( \| \cdot \| \), then both (16) with \( b \leftarrow b / m \), and (17) with \( b \leftarrow b/2 \) and h=1 hold true.
- If (16), there exists a subfamily \( {\mathcal F}’ \subset {\mathcal F} \) satisfying \( \| {\mathcal F}’ \| > 2^{-1} \| {\mathcal F} \| \) and the \( \Gamma \left( b / 6 \right) \)-condition on \( \| \cdot \| \).
- If (17) for a given \( g>1 \), \( b< m^2\), and \( h=1 \), the family \( {\mathcal F} \) satisfies the \( \Gamma \left( b m^{-3/g} \right) \)-condition on \( \| \cdot \| \).
It was given last time essentially saying the three \( \Gamma \)-conditions are similar depending on the choice of b, h and \( \gamma \). Let’s finish justifying it.
Prop. 1-3 is found true as follows. If \( {\mathcal F} \) doesn’t satisfy the \( \Gamma \left( b m^{-3/g} \right) \)-condition on \( \| \cdot \| \), there exists a nonempty set T of cardinality u such that \( \| {\mathcal F}[T] \| \ge ( b m^{-3/g} )^{-u} \| {\mathcal F} \| \), so $$
\sum_{T \in {X \choose u}} | {\mathcal F}[T] |^g > \left( \frac{b}{m^{3/g}} \right)^{-ug} | {\mathcal F} |^g > b^{-u(g-1)} m^u | {\mathcal F} |^g > b^{-u(g-1)} {m \choose u} | {\mathcal F} |^g,
$$ whose second inequality is due to \( b< m^2 \). This falsifies (18) last time, and also (17), impossible to hold true. Hence (17) means the \( \Gamma \)-condition of \( {\mathcal F} \).
To show Prop. 1-2, we want to construct a large enough subfamily \( {\mathcal F}’ \subset {\mathcal F} \) satisfying the \( \Gamma(b/6) \)-condition from (16). Copy \( {\mathcal F} \) into \({\mathcal F}’ \) and try u=|T|=1 first: find a singleton set \( T \in {X \choose u} \) such that \( \| {\mathcal F}'[T] \| \ge (b/3)^{-u} \| {\mathcal F} \| \). If found, delete the \( {\mathcal F}'[T] \) from \( {\mathcal F}’ \). Continue until such a T is no longer detected.
This reduces less than \( 3^{-u} \) of the total norm \( \| {\mathcal F} \| \), otherwise it finds such a u-set \( \frac{3^{-u} | {\mathcal F} | }{(b/3)^{-u} | {\mathcal F} |} = (b/9)^u \) times or less to cause this contradiction: $$ \sum_{T \in {X \choose u}} \| {\mathcal F}[T] \|^2 \ge \left( \frac{b}{9} \right)^u \left[ \frac{\left\| {\mathcal F} \right\|}{(b/3)^u} \right]^2 = b^{-u}\left\| {\mathcal F} \right \|^2, $$ falsifying (16). The inequality in the middle is exactly seen by the optimization problem (10) we talked about in the 9/17 post. It’s true because the solution to (10) is \( A^2 / N \).
Do the same for u=2, 3, 4, \( \ldots \), for each of which we eliminate less than \( 3^{-u} \) of \( \| {\mathcal F} \|. \) In the end we have the remaining \( 1 – 3^{-1} – 3^{-2} – \cdots > \frac{1}{2} \) with no T meeting \( \| {\mathcal F}’ [T] \| \ge (b/3)^{-|T|} \| {\mathcal F} \| \). Therefore, the obtained \( {\mathcal F}’ \) is more than a half of \( {\mathcal F} \) satisfying the \( \Gamma (b/6) \)-condition, finishing the proof.
We now notice a fact motivating the rank-g construction mentioned last time. If we use (16) with the collective \( \Gamma(b) \)-condition of rank 2, the b is actually equivalent to bm for the basic \( \Gamma \)-condition as seen in Prop. 1-1. If we can use (17) in a proof, the gap between the two is about \( m^{3/g} \) or less, which is a lot smaller than m if g is a sufficiently large constant. The main result of Paper 3 is \( \Gamma \left( m^{\frac{1}{2} + \epsilon} \right) \) meaning three mutually disjoint sets in \( {\mathcal F} \) as has been said. That \( \epsilon \) comes from the \( 3/g \) here.
Additionally, (16) with \( b = 5 \gamma m n / l \) alone means the claims of the double mark lemma and Theorem 1.2 of Paper S3 in the weighted case. You can check it with the same arguments in the 8/18 post, where we see $$ \sum_{T \in {X \choose j}} \|{\mathcal F}[T] \|^2 < b^{-j} \| {\mathcal F} \|^2, \qquad \Rightarrow \qquad \| {\mathcal P}_j \| \le \left( \frac{m}{b} \right)^j \| {\mathcal F} \|^2. $$ Apply this to the discussion in the 9/2 post and others.
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