The following trees represent ways to assign primes in a completely multiplicative sequence. If a leaf is labeled {N,k}, it means that the corresponding assignment has with absolute value “drift”.

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In the Erdos Homogeneous Discrepancy problem, we are interested not in subwords, but in the pattern of pluses and minuses that are along HAPs. This image is the tree of patterns that we see in a particular 1124-term sequence that has shockingly small discrepancy. Each vertex label includes the number of HAPs that begin with that particular pattern. Here’s the same tree for the subsequences RauzyTreeStub1124, RauzyTreeStub1124d2, RauzyTreeStub1124d3, RauzyTreeStub1124d4, RauzyTreeStub1124d6, and RauzyTreeStub1124d8. A word of caution: the subsequences are shorter than the original, and so the tree represents less data. The fact that the tree appears cleaner could be illusory.

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Give it a try, and let me know how it turns out for you. I’ll do the same in this space.

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