Experimental results

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Perhaps we should have two kinds of subpages to this page: Pages about finding examples, and pages about analyzing them?

Experimental data

• The first 1124-sequence with discrepancy 2. Some more description
• Other length 1124 sequences with discrepancy 2. Some more description
• Some data about the problem with different upper and lower bound. Some more description
• Sequences taking values in [math]\displaystyle{ \mathbb{T} }[/math]:
  • 4th roots of unity
  • 6th roots of unity
• A sequence of length 407 with discrepancy 2 such that [math]\displaystyle{ x_n=x_{32 n} }[/math] for every n.
• More T32-invariant sequences.
• Long multiplicative sequences.

Wish list

• Find long/longest quasi-multiplicative sequences with some fixed group G, function [math]\displaystyle{ G\to \{-1,1\} }[/math] and maximal discrepancy C
  • [math]\displaystyle{ G=C_6 }[/math] and the function that sends 0,1 and 2 to 1 (because this seems to be a good choice)
• Do a "Mark-Bennet-style analysis" of one of the new 1124-sequences. [1] Also done (by Mark Bennet).
• . Take a moderately large k and search for the longest sequence of discrepancy 2 that's constructed as follows. First, pick a completely multiplicative function f to the group [math]\displaystyle{ C_{2k} }[/math]. Then set [math]\displaystyle{ x_n }[/math] to be 1 if f(n) lies between 0 and k-1, and -1 if f(n) lies between k and 2k-1. Alec has already done this for k=1 and partially done it for k=3.
• Search for the longest sequence of discrepancy 2 with the property that [math]\displaystyle{ x_n=x_{32n} }[/math] for every n. The motivation for this is to produce a fundamentally different class of examples (different because their group structure would include an element of order 5). It's not clear that it will work, since 32 is a fairly large number. However, if you've chosen [math]\displaystyle{ x_{32n} }[/math] then that will have some influence on several other choices, such as [math]\displaystyle{ x_{4n},x_{8n} }[/math] and [math]\displaystyle{ x_{16n} }[/math], so maybe it will lead to something interesting. Alec has made a start on this and an initial investigation suggests that the sequence he has found does indeed have some [math]\displaystyle{ C_{10} }[/math]-related structure.

• ... you are welcome to add more.