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Results tagged with discrepancy-theory
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user 2480
10
votes
1 rectangle <= 4 squares
The upper bound is <3.95.
I hope the code below will show correctly...
It proves that assuming a sum >=3.95 in the central AxB rectangle of the grid
({-B,-B+A,-2A,-A,0,A,2A,B-A,B}+{0,A}) x ({-2B,-B- …
- 5,103
4
votes
1 rectangle <= 4 squares
Here is a summary for the $\mathbb{R}^2$ situation.
Upper limit: no improvement over the 3.8 known on $\mathbb{Z}^2$.
To create specific examples I modified the $\mathbb{Z}^2$ ones by uniformely
spr …
3
votes
1 rectangle <= 4 squares
There is a new upper bound of 254/67 (= 3.79104477...).
Define 6 sets of cardinality 4:
X1={-B+A, 0, A, B}
Y1={0, A, B-A, B}
X2={-B, -B+3A, B-2A, B+A}
Y2={-2B+A, -A, B+A, 3B-A}
X3={-6B+2A, -2B-2A, …
42
votes
8
answers
4k
views
1 rectangle <= 4 squares
Almost 25 years ago a professor at Indiana U showed me the following problem:
given a map $\mathbb{Z}^2\rightarrow\mathbb{R}$ such that the sum inside every square (parallel to the axes) is $\leq1$ i …