diff --git a/constants/10c.md b/constants/10c.md index 653f11d..1ba449e 100644 --- a/constants/10c.md +++ b/constants/10c.md @@ -2,15 +2,32 @@ ## Description of constant -$C_{10c}$ is the least constant $K$ for which one has -$$\mathrm{disc}(A) \le K\sqrt{n}\qquad\text{for all }n\text{ and all }A\in[-1,1]^{n\times n}. +$C\_{10c}$ is the least constant $K$ for which one has + +$$ +\mathrm{disc}(A) \le K\sqrt{n} +\qquad\text{for all }n\text{ and all }A\in[-1,1]^{n\times n}. +$$ + +Here the **discrepancy** $\mathrm{disc}(A)$ is defined as + +$$ +\mathrm{disc}(A) := \min_{x\in\{\pm 1\}^n} \|Ax\|_\infty. +$$ + +Equivalently, if the linear forms are + +$$ +L_i(x_1,\dots,x_n)=\sum_{j=1}^n a_{ij}x_j, $$ -where the **discrepancy** $\mathrm{disc}(A)$ is defined as -$$\mathrm{disc}(A) \;:=\; \min_{x\in\{\pm 1\}^n}\ \|Ax\|_\infty.$$ -Equivalently, if $L_i(x_1,\dots,x_n)=\sum_{j=1}^n a_{ij}x_j$ are $n$ linear forms, + then -$$\mathrm{disc}((a_{ij})_{i,j=1}^n)=\min_{\varepsilon\in\{\pm 1\}^n}\max_{1\le i\le n}|L_i(\varepsilon)|.$$ +$$ +\mathrm{disc}((a_{ij})_{i,j=1}^n)= +\min_{\varepsilon\in\{\pm 1\}^n} +\max_{1\le i\le n}\lvert L_i(\varepsilon)\rvert. +$$ ## Known upper bounds @@ -26,20 +43,123 @@ $$\mathrm{disc}((a_{ij})_{i,j=1}^n)=\min_{\varepsilon\in\{\pm 1\}^n}\max_{1\le i | Bound | Reference | Comments | | ---: | :--- | :--- | | $1$ | Trivial | $A=[1]$. Also achieved by Hadamard matrices [Band2024]. | -| $\sqrt{2}\approx 1.414214$ | [Band2024] | $A = \begin{bmatrix}1&1\\1&-1\end{bmatrix}$. | +| $\sqrt{2}\approx 1.414214$ | [Band2024] | The 2 by 2 sign matrix with rows $(1,1)$ and $(1,-1)$. | +| $4/\sqrt{6}\approx 1.632993$ | [G2026] | A 6 by 6 sign matrix with exact discrepancy $4$. | + +## Certificate for the $4/\sqrt{6}$ lower bound + +Use the following sign matrix: + +```text + 1 1 1 1 1 1 +-1 -1 -1 1 1 1 +-1 1 1 -1 1 1 +-1 1 -1 1 -1 1 + 1 1 -1 -1 1 1 + 1 -1 1 -1 -1 1 +``` + +All entries are `±1`. For a sign vector `x`, let `S` be the set of columns where the corresponding coordinate of `x` is `-1`. For row number $i$, let `T_i` be the set of columns where that row has entry `-1`: + +```text +T1 = {} +T2 = {1,2,3} +T3 = {1,4} +T4 = {1,3,5} +T5 = {3,4} +T6 = {2,4,5} +``` + +Then + +$$ +(Ax)_i = 6 - 2\lvert S\triangle T_i\rvert. +$$ + +Therefore the absolute value of the $i$th coordinate of `Ax` is at least $4$ whenever `S` is within Hamming distance $1$ of either `T_i` or the complement of `T_i`. The relevant centers are + +```text +{}, 123, 14, 135, 34, 245, 123456, 456, 2356, 246, 1256, 136 +``` + +where, for example, `123` denotes the set `{1,2,3}`. + +The sets of size 0, 1, 5, and 6 are covered by the centers `{}` and `123456`. The sets of size 2 are covered as follows: + +| S | center | +| :---: | :---: | +| 12 | 123 | +| 13 | 123 | +| 14 | 14 | +| 15 | 135 | +| 16 | 136 | +| 23 | 123 | +| 24 | 245 | +| 25 | 245 | +| 26 | 246 | +| 34 | 34 | +| 35 | 135 | +| 36 | 136 | +| 45 | 456 | +| 46 | 456 | +| 56 | 456 | + +Since the centers are closed under complementation, this also covers the sets of size 4. The sets of size 3 are covered as follows: + +| S | center | +| :---: | :---: | +| 123 | 123 | +| 124 | 14 | +| 125 | 1256 | +| 126 | 1256 | +| 134 | 14 | +| 135 | 135 | +| 136 | 136 | +| 145 | 14 | +| 146 | 14 | +| 156 | 1256 | +| 234 | 34 | +| 235 | 2356 | +| 236 | 2356 | +| 245 | 245 | +| 246 | 246 | +| 256 | 1256 | +| 345 | 34 | +| 346 | 34 | +| 356 | 2356 | +| 456 | 456 | + +Thus every sign vector satisfies + +$$ +\|Ax\|_\infty \ge 4. +$$ + +Equality is attained. For example, + +```text +x = (1,-1,1,1,1,1)^T +Ax = (4,2,0,-2,0,2)^T +``` + +Hence $\mathrm{disc}(A)=4$, and consequently + +$$ +C_{10c}\ge \frac{4}{\sqrt{6}}\approx 1.632993. +$$ ## Further remarks - For large $n$, the best asymptotic lower bound remains $1$ [Band2024]. -- Replacing the entrywise bound $|a_{ij}|\le 1$ by an $\ell_2$-bound on columns - leads to the Komlós conjecture, which would imply (after scaling) Spencer-type discrepancy bounds. +- Replacing the entrywise bound by an $\ell\_2$-bound on columns leads to the Komlós conjecture, which would imply, after scaling, Spencer-type discrepancy bounds. ## References - [AS2008] Alon, N.; Spencer, J. *The Probabilistic Method*, 3rd ed. Wiley, 2008. (See the discussion around “Six Standard Deviations Suffice”.) -- [Band2024] Bandeira, A. S. [*Did just a couple of deviations suffice all along?](https://randomstrasse101.math.ethz.ch/posts/HowManyDeviations/) (problems 10–14).* Randomstrasse 101 blog post (Dec 19, 2024). +- [Band2024] Bandeira, A. S. [*Did just a couple of deviations suffice all along?*](https://randomstrasse101.math.ethz.ch/posts/HowManyDeviations/) (problems 10–14). Randomstrasse 101 blog post (Dec 19, 2024). - [Ban2010] Bansal, N. *Constructive algorithms for discrepancy minimization.* FOCS 2010, 3–10. - [Bel2013] Belshaw, A. W. *Strong Normality, Modular Normality, and Flat Polynomials: Applications of Probability in Number Theory and Analysis.* PhD thesis, Simon Fraser University, 2013. +- [G2026] Griego, Sebastian. 6 by 6 sign-matrix certificate for C10c, submitted to this repository (2026). - [LM2015] Lovett, S.; Meka, R. *Constructive discrepancy minimization by walking on the edges.* SIAM J. Comput. **44** (5) (2015), 1573–1582. [arXiv:1203.5747](https://arxiv.org/abs/1203.5747) - [MO175826] MathOverflow. [*Spencer’s “six standard deviations” theorem – better constants?*](https://mathoverflow.net/questions/175826/) Question 175826 (2014). - [PV2022] Pesenti, L.; Vladu, A. *Discrepancy Minimization via Regularization.* [arXiv:2211.05509](https://arxiv.org/abs/2211.05509)