Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

Abstract

Given a set system $(X, \mathcal {R})$ such that every pair of sets in $\mathcal {R}$ have large symmetric difference, the Shallow Packing Lemma gives an upper bound on $|\mathcal {R}|$ as a function of the shallow-cell complexity of $\mathcal {R}$. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system $(X, \mathcal {R})$ with shallow-cell complexity $\varphi (\cdot , \cdot )$ and a parameter $\epsilon > 0$, there exists a collection, called an $\epsilon$-Mnet, consisting of $O\bigl ( \frac{1}{\epsilon } \,\varphi \bigl ( O\bigl (\frac{1}{\epsilon } \bigr ), O(1)\bigr ) \bigr )$ subsets of X, each of size $\Omega ( \epsilon |X| )$, such that any $R \in \mathcal {R}$ with $|R| \ge \epsilon |X|$ contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal $\epsilon$-net bound follows.

Appendix A: Generalization of the Packing Lemma

A set system $(X, \mathcal {R})$ is an l-wise k-shallow $\delta$-packing if $|R| \le k$ for all $R \in \mathcal {R}$, and further for all distinct $A_{1}, \dots , A_{l} \in \mathcal {R}$, we have

$$\begin{aligned} \left| \left( A_{1} \cup \dots \cup A_{l} \right) {\big \backslash } \left( A_{1} \cap \dots \cap A_{l} \right) \right| \ge \delta . \end{aligned}$$

A routine generalization of the proof in [21, 25] leads to the following.

Theorem A.1

(l-Wise k-Shallow $\delta$-Packing Lemma) Let $(X, \mathcal {R})$ be a set system with $|X| = n$. Let $d, \, k, \, l, \, \delta > 0$ be four integers such that ${{\,\mathrm{\textsc {VC{-}dim}}\,}}(\mathcal {R}) \le d$, and $\mathcal {R}$ is an l-wise k-shallow $\delta$-packing. If $\mathcal {R}$ has shallow-cell complexity $\varphi _{\mathcal {R}}\left( \cdot , \cdot \right)$, then

$$\begin{aligned} |\mathcal {R}| = O\left( \frac{l^3 d n}{\delta } \cdot \varphi _{\mathcal {R}} \left( \frac{8dl^2n}{\delta }, \frac{32 d k l^3}{\delta } \right) \right) . \end{aligned}$$

Remark A.2

The above result implies Haussler’s Packing Lemma (set $l=2, k=n$), the Shallow Packing Lemma 1.2 (set $l=2$) and the result of Fox et al. [16, Lem. 2.5] (set $k=n$).

The proof follows by combining the ideas in [16, 21, 25].

Lemma A.3

Let $(X, \mathcal {R})$ be a set system with $|X| = n$. Let d, l, $\delta$ be three integers such that ${{\,\mathrm{\textsc {VC{-}dim}}\,}}(\mathcal {R})\le d$, and $\mathcal {R}$ is an l-wise $\delta$-packing. If $A \subseteq X$ is a uniformly selected random sample of size $\frac{8l(l-1)dn}{\delta } -1$, then $|\mathcal {R}| \le 2l \cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}|_{A}|\right]$.

Proof

Pick a random sample R of size $s = \frac{8l(l-1)dn}{\delta }$ from X. Let $G_R = (\mathcal {R}|_{R}, E_{\mathcal {R}})$ be the unit distance graph on $\mathcal {R}|_{R}$, with an edge between any two sets whose symmetric difference is a singleton. Define the weight of a set $S' \in \mathcal {R}|_{R}$ to be the number of sets of $\mathcal {R}$ whose projection in $\mathcal {R}|_R$ is $S'$, i.e. $w(S') = |\{ r\in \mathcal {R}\ |\ r\cap R = S' \} |$. Define the weight of an edge $\{S'_i, S'_j\} \in E_{\mathcal {R}}$ as $w(S'_i, S'_j) = \min \{ w(S'_i), w(S'_j)\}$. Let $W := \sum _{e \in E_{\mathcal {R}}} w(e)$. $\square$

We use the following result from [21, Chap. 5, Proof 5.14].

Claim A.4

[21, Proof 5.14 from Chap. 5] $W \le 2d \cdot |\mathcal {R}|$.

Pick R by first picking a set A of $s-1$ elements and then selecting the remaining element a uniformly from $X {\setminus } A$. Let $W_1$ be the weight of the edges in $G_R$ for which the element a is the symmetric difference. By symmetry, we have ${{\,\mathrm{\mathbb {E}}\,}}\left[ W\right] = s \cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ W_1\right]$.

We use the following lower bound on the conditional expectation of $W_1$ with respect to A.

Claim A.5

${{\,\mathrm{\mathbb {E}}\,}}\left[ W_1 | A \right] \ge \frac{\delta / n}{2l(l-1)} \Big (|\mathcal {R}| - l \, |\mathcal {R}|_{A}| \Big )$.

The proof of this claim is given at the end of this section.

Using the fact that ${{\,\mathrm{\mathbb {E}}\,}}\left[ W\right] = s \cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ W_1\right]$, one can compute an upper bound on ${{\,\mathrm{\mathbb {E}}\,}}\left[ W\right]$:

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left[ W\right] = s \cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ W_1\right]&= s \cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ {{\,\mathrm{\mathbb {E}}\,}}[W_1|A]\right] \\&\ge s {{\,\mathrm{\mathbb {E}}\,}}\left[ \frac{\delta }{2l(l-1)n} \Big (|\mathcal {R}| - l \, |\mathcal {R}|_{A}| \Big )\right] \text { (by Claim A.5)} \\&= 4d {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}| - l \, |\mathcal {R}|_{A}|\right] \\&= 4d\bigl (|\mathcal {R}| - l {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}|_{A}|\right] \bigr ). \end{aligned}$$

Combining Claim A.4 and the above lower bound on ${{\,\mathrm{\mathbb {E}}\,}}\left[ W\right]$, we get

$$\begin{aligned} 2d |\mathcal {R}| \ge {{\,\mathrm{\mathbb {E}}\,}}\left[ W\right] \ge 4d |\mathcal {R}| - 4dl\cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}|_{A}|\right] . \end{aligned}$$

This implies $|\mathcal {R}| \le 2l\cdot {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}|_{A}|\right]$. $\square$

Proof of Theorem A.1

Let $A \subseteq X$ be a random sample of size $s : = \frac{8l(l-1)dn}{\delta }-1$. Let $\mathcal {R}_1 = \left\{ S \in \mathcal {R}\text { s.t. } |S \cap A | \ge 4l \cdot \frac{ks}{n} \right\}$.

Each element $x \in X$ belongs to A with probability $\frac{s}{n}$, and thus the expected number of elements in A from a fixed set of t elements is $\frac{ts}{n}$. This implies that ${{\,\mathrm{\mathbb {E}}\,}}\left[ |S \cap A |\right] \le \frac{k s}{n}$ as $|S| \le k$ for all $S \in \mathcal {R}$. Markov’s inequality then bounds the probability of a set of $\mathcal {R}$ belonging to $\mathcal {R}_1$:

$$\begin{aligned} \Pr \left[ S \in \mathcal {R}_1\right] = \Pr \left[ |S \cap A | \ge 4l \cdot \frac{k s}{n} \right] \le \frac{1}{4l}. \end{aligned}$$

Thus

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}|_{A }|\right] \le {{\,\mathrm{\mathbb {E}}\,}}\left[ |\mathcal {R}_1|\right] + {{\,\mathrm{\mathbb {E}}\,}}\left[ |(\mathcal {R}{\setminus } \mathcal {R}_1)|_{A }|\right]\le & {} \sum _{S \in \mathcal {R}} \Pr \left[ S \in \mathcal {R}_1 \right] + s \, \cdot \varphi \Big ( s, 4l\cdot \frac{k s}{n} \Big )\\\le & {} \frac{|\mathcal {R}|}{4l} + s \cdot \varphi \Big ( s , 4l \cdot \frac{k s}{n} \Big ), \end{aligned}$$

where we used the fact that $|(\mathcal {R}{\setminus } \mathcal {R}_1)|_{A }| \le |A| \cdot \varphi (|A|, t)$, where $t = \max _{S \in \mathcal {R}{\setminus } \mathcal {R}_1} |S| < 4l \frac{ks}{n}$.

Now the bound follows from Lemma A.3. $\square$

Finally we give the proof of Claim A.5.

Proof of Claim A.5

Consider a set $Q \in \mathcal {R}|_{A}$, and let $\mathcal {R}_Q$ be the sets of $\mathcal {R}$ whose projection is Q. Once the choice of a has been made, Q will be split into two sets, those sets containing that choice of a – say there are $b_1$ of these, and those sets not containing a, say a number $b_2$. From the definition of weights, the expected contribution of sets of $\mathcal {R}_Q$ to edge weight will be ${{\,\mathrm{\mathbb {E}}\,}}\left[ \min \{b_1, b_2\} \right] \ge \frac{{{\,\mathrm{\mathbb {E}}\,}}\left[ b_1 b_2 \right] }{b_1+b_2}$. The above inequality follows from the fact $\min \{b_1 , b_2 \} \ge \frac{b_1 b_2}{b_1 + b_2}$. Observe that $b_1 b_2$ is the number of ordered pairs $(S_1 , S_2 ) \in \mathcal {R}_Q \times \mathcal {R}_Q$ with $a \in S_1$ and $a \not \in S_2$. Therefore for each fixed pair of sets $(S_1 , S_2) \in \mathcal {R}_Q \times \mathcal {R}_Q$, the probability that the randomly chosen last element $a \in S_1 {\setminus } S_2$ is $\frac{|S_1 {\setminus } S_2 |}{n-s-1}$. Therefore the contribution of $(S_1 , S_2)$ in $\mathcal {R}_Q$ to $b_1b_2$ is $\frac{|S_1 {\setminus } S_2 |}{n-s-1}$. Noting that $b = b_1+b_2$ is fixed independent of the choice of a, summing up over all pairs of sets in $\mathcal {R}_Q$, we get the expected contribution of the sets in $\mathcal {R}_Q$ to the edge weight to be at least

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left[ \min \{b_1, b_2\} \right]&\ge \frac{{{\,\mathrm{\mathbb {E}}\,}}\left[ b_1 b_2 \right] }{b_1 + b_2 } \\&\ge \frac{1}{b_1 + b_2 } \Bigg ( \sum _{(S_1,\, S_2 )\, \in \mathcal {R}_Q \times \mathcal {R}_Q} \Pr \left[ a \in S_1 {\setminus } S_2 \right] \Bigg )\\&\ge \frac{1}{b_1 + b_2 } \Bigg ( \sum _{ S_1 , \, S_2 \, (\ne S_1 ) \, \in \mathcal {R}_Q } \Pr \left[ a \in S_1 {\setminus } S_2 \right] + \Pr \left[ a \in S_2 {\setminus } S_1 \right] \Bigg )\\&= \frac{1}{b_1 + b_2 } \Bigg ( \sum _{ S_1 ,\, S_2 \, (\ne S_1 ) \,\in \mathcal {R}_Q } \Pr \left[ a \in S_1 \mathbin {\Delta }S_2 \right] \Bigg )\\&= \frac{1}{b_1 + b_2 } \Bigg ( \sum _{S_1 ,\, S_2 \, (\ne S_1 ) \, \in \mathcal {R}_Q } \frac{|S_1 \mathbin {\Delta }S_2 |}{n-s+1} \Bigg ). \end{aligned}$$

For all l sets $S_1 , \dots , S_l \in \mathcal {R}_Q$, we have

$$\begin{aligned} \bigcup _{2 \le j \le l} S_1 \mathbin {\Delta }S_j = \left( S_1 \cup \dots \cup S_l \right) {\setminus } \left( S_1 \cap \dots \cap S_l \right) . \end{aligned}$$

And since $\mathcal {R}$ is an l-wise $\delta$-packing we get

$$\begin{aligned} \sum _{2 \le j \le l} |S_1 \mathbin {\Delta }S_j| \ge \left| \left( S_1 \cup \dots \cup S_l \right) {\setminus } \left( S_1 \cap \dots \cap S_l \right) \right| \ge \delta . \end{aligned}$$

So for every l tuple there exists one pair $(S_1 , S_j )$ with $| S_1 \mathbin {\Delta }S_j | \ge \frac{\delta }{l-1}$. Define the graph $G\left[ \mathcal {R}_Q \right] : = (\mathcal {R}_Q , E_Q )$, where $\{S_1 , S_2\} \in E$ if $|S_1 \mathbin {\Delta }S_2 | \ge \frac{\delta }{l-1}$. As $\mathcal {R}_Q$ is an l-wise $\delta$-packing we do not have independent sets of size l in $G\left[ \mathcal {R}_Q \right]$. From Turán’s theorem, see [28], we have $|E_Q | \ge \frac{b(b-l)}{2l}$.

Therefore

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left[ \min \{b_1, b_2\} \right]&\ge \frac{1}{b} \Bigg ( \sum _{S_1 , \, S_2 \, (\ne S_1 ) \, \in \mathcal {R}_Q } \frac{|S_1 \mathbin {\Delta }S_2 |}{n-s+1} \Bigg )\\&\ge \frac{1}{b} \Bigg ( \sum _{ \left\{ S_1 , \, S_2 \right\} \in E_Q } \frac{|S_1 \mathbin {\Delta }S_2 |}{n-s+1} \Bigg )\\&\ge \frac{|E_Q |}{b} \cdot \frac{(\delta / n)}{l-1} \\&\ge \frac{(\delta / n )}{2l(l-1)} \cdot \left( | \mathcal {R}_Q | - l \right) . \end{aligned}$$

The last inequality follows from the facts $|E_Q | \ge \dfrac{b(b-l)}{2l}$ and $|\mathcal {R}_Q | = b$.

Summing up over all sets of $\mathcal {R}|_{A}$,

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}\left[ W_1 | A \right] \ge \frac{1}{2l(l-1)} \sum _{Q \in \mathcal {R}|_{A}} \frac{\delta }{n} \Big (|\mathcal {R}_{Q}|- l \Big ) = \frac{\delta / n}{2l (l-1)} \Big ( |\mathcal {R}| - l \, |\mathcal {R}|_{A}|\Big ). \end{aligned}$$

$\square$