We’ve finished the proof of Lemma 1 posed on 5/13/24. Let’s call it \(r\)-split lemma in this blog. As said, a set \( S \) is on an \(r\)-split \( {\boldsymbol X} \) if it satisfies Condition \( r \) given last time. Denoting the family of all \( m \)-sets on \( {\boldsymbol X} \) by \( {{\boldsymbol X} \choose m } \), let’s see some additional facts related to it.
Remark 1. The lemma can be weighted with primitive norm \( \| \cdot \| \): as on 5/13/24 and 1/15/24, primitive norm is the simplest norm defined by a one-parameter weight function \(w_* : 2^X \rightarrow {\mathbb R}_{\ge 0} \) we dealt with a lot discussing collective \( \Gamma \)-conditions. In Paper 3, the \(r\)-split lemma with such weight is stated as Corollary 3.2 on p. 11, which is rephrased into:
Corollary 3.2: in an \( X \) primitively weighted with \( \| \cdot \| \), a family \( {\mathcal F} \subset {X \choose m} \) meets \[ \left \| {\mathcal F} \cap {{\boldsymbol X} \choose m} \right \| \ge {n/r \choose m/r}^ r \frac{ \| {\mathcal F} \|}{{n \choose m}}, \] for any \( r \in [n] \) that divides both \( n \) and \(m \), and some \(r \)-split \( {\boldsymbol X} \).
We did the same generalization for the EGT on 10/8/23 and after. By the update \( | \cdot | \rightarrow \| \cdot \| \), it changed the sparsity into \( \kappa ({\mathcal F} ) = \ln {n \choose m} – \ln \| {\mathcal F} \| \) that worked for the proof steps for the weighted EGT as well. Almost nothing else was necessary to change.
The update \( | \cdot | \rightarrow \| \cdot \| \) also works for the \(r\)-split lemma to prove the corollary. It is straightforward to check this in our proof last time.
Remark 2. The lower bound on the RHS can be further bounded by \( \|{\mathcal F} \| \exp \left[ – \frac{r}{2} \ln 2 \pi \frac{m}{r} – O\left( \frac{r^2}{m} \right) \right] \): the original split lemma says \( \left| {\mathcal F} \cap {{\boldsymbol X} \choose m} \right| \ge \left( \frac{n}{m} \right)^m |{\mathcal F}| {n \choose m}^{-1} > |{\mathcal F}| e^{-m} \), or \( \kappa \left[ {\mathcal F} \cap {{\boldsymbol X} \choose m} \right] < m \). We can find a similar sparsity bound for an \(r\)-split \( {\boldsymbol X} \) by Lemma 1.1 of Paper S3: we used it on 9/9/23 to prove the inequality (3) on 8/4/23.
We’ll see other remarks in the next couple of posts.
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