Suppose a school purchases a set $\mathcal{S}$ of balls, say

$$\displaystyle \mathcal{S} = \{b_1, b_2, \cdots, b_n\}$$

with $n$ very large. The balls $b_j$ are pairwise distinct and have distinct weights; by abuse of notation, we simply say that $b_j$ is the real number representing the weight of the $j$-th ball, normalized so that

$$\displaystyle \sum_{j=1}^n b_j = 1.$$

Further assume that the weights of the balls are such that for any two distinct, disjoint subcollections of balls $\mathcal{S_1}, \mathcal{S}_2$ the total weights of the balls in $\mathcal{S_1}$ and $\mathcal{S}_2$ are different.

On each day, the balls are given to the children to play. At the end of each day, the custodian collects the balls and put them in one of three bins. The custodian does not care about which ball goes where, but does care to make sure that the total weights in each of the bins is approximately the same. He does not use a scale and can only approximate the weights by hand. In other words, he ensures that the total weight of the balls in each bin lies in the interval $(1/3 - \kappa, 1/3 + \kappa)$ for some predetermined $\kappa$, say $0 < \kappa < 1/100$.

The custodian also records six triples at the end of each day, corresponding to the set of balls in each bin and its permutations. In other words he records the triple

$$\displaystyle \left(\mathcal{S}_1^{(k)}, \mathcal{S}_2^{(k)}, \mathcal{S}_3^{(k)} \right)$$

as well as its permutations and the end of the $k$-th day.

Out of curiosity, the custodian computes, at the end of each day, the sums

$$\displaystyle \sum_{j=1}^3 \sum_{\sigma \in \mathfrak{S}_3} w \left(\mathcal{S}_j^{(k)} \cap \mathcal{S}_{\sigma(j)}^{(k^\prime)} \right)$$

for $0 \leq k^\prime \leq k$, $\mathfrak{S}_3$ is the symmetric group on three letters, and for a subset $\mathcal{B}$ of $\mathcal{S}$ we have

$$\displaystyle w(\mathcal{B}) = \sum_{b \in \mathcal{B}} b.$$

For a positive number $0 < \delta < 1$ let $m(\mathcal{S}, \delta)$ be the smallest positive integer $k$ such that

$$\displaystyle \max_{0 \leq k^\prime \leq k} \sum_{j=1}^3 \sum_{\sigma \in \mathfrak{S}_3} w \left(\mathcal{S}_j^{(k)} \cap \mathcal{S}_{\sigma(j)}^{(k^\prime)} \right) \geq 1 - \delta.$$

Is there a good estimate for $m(\mathcal{S}, \delta)$ in terms of $n = |\mathcal{S}|$ and $\delta$ only?

Suppose a school purchases a set $\mathcal{S}$ of balls, say

$$\displaystyle \mathcal{S} = \{b_1, b_2, \cdots, b_n\}$$

with $n$ very large. The balls $b_j$ are pairwise distinct and have distinct weights; by abuse of notation, we simply say that $b_j$ is the real number representing the weight of the $j$-th ball, normalized so that

$$\displaystyle \sum_{j=1}^n b_j = 1.$$

Further assume that the weights of the balls are such that for any two distinct, disjoint subcollections of balls $\mathcal{S_1}, \mathcal{S}_2$ the total weights of the balls in $\mathcal{S_1}$ and $\mathcal{S}_2$ are different. We also fix a positive number $0 < \delta < 1/3$.

On each day, the balls are given to the children to play. At the end of each day, the custodian collects the balls and put them in one of three bins. The custodian does not care about which ball goes where, but does care to make sure that the total weights in each of the bins is not too large. He does not use a scale and can only approximate the weights by hand. In other words, he ensures that the total weight of the balls in each bin lies in the interval $(0, (1- \delta)/2)$.

The custodian also records six triples at the end of each day, corresponding to the set of balls in each bin and its permutations. In other words he records the triple

$$\displaystyle \left(\mathcal{S}_1^{(k)}, \mathcal{S}_2^{(k)}, \mathcal{S}_3^{(k)} \right)$$

as well as its permutations and the end of the $k$-th day.

Out of curiosity, the custodian computes, at the end of each day, the sums

$$\displaystyle \sum_{j=1}^3 \sum_{\sigma \in \mathfrak{S}_3} w \left(\mathcal{S}_j^{(k)} \cap \mathcal{S}_{\sigma(j)}^{(k^\prime)} \right)$$

for $0 \leq k^\prime \leq k$, $\mathfrak{S}_3$ is the symmetric group on three letters, and for a subset $\mathcal{B}$ of $\mathcal{S}$ we have

$$\displaystyle w(\mathcal{B}) = \sum_{b \in \mathcal{B}} b.$$

Let $m(\mathcal{S}, \delta)$ be the smallest positive integer $k$ such that

$$\displaystyle \max_{0 \leq k^\prime \leq k} \sum_{j=1}^3 \sum_{\sigma \in \mathfrak{S}_3} w \left(\mathcal{S}_j^{(k)} \cap \mathcal{S}_{\sigma(j)}^{(k^\prime)} \right) \geq 1 - \delta.$$

Is there a good estimate for $m(\mathcal{S}, \delta)$ in terms of $n = |\mathcal{S}|$ and $\delta$ only?