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My interpretation in layman's terms of what this paper has proved - Take an infinite sequence of +1 and -1, for example 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, ...
You get an evenly spaced subsequence by starting on the n'th element, and picking every n'th element after that (and stopping after finitely many terms). So for example, we could pick every 2nd element of this sequence and get 1 (stopping after 1 term)
1, 1 (stopping after 2 terms)
1, 1, -1 (stopping after 3 terms)
1, 1, -1, 1 (stopping after 4 terms)
1, 1, -1, 1, -1 (stopping after 5 terms)
The discrepancy of an evenly spaced subsequence is obtained by adding together all the members of the subsequence and taking the absolute value. So the discrepancies of the sequences above are 1, 2, 1, 2, and 1.The challenge is to find an evenly-spaced subsequence with as large a discrepancy as possible. For example, in a given sequence, can you find an evenly-spaced subsequence with a discrepancy of 10? of 100? of a million? The paper has (apparently) proved that for any sequence, there is no upper limit to the discrepancies of evenly spaced subsequences, i.e. no matter how large the discrepancy of a subsequence you have found, there is always one larger. The amazing thing about this result, to me, is the fact that it holds for any sequence of +1 and -1. Even if you try to engineer a sequence whose subsequences all have very small discrepancy, in some sense "there isn't enough room". You are always doomed to come up with a sequence containing evenly-spaced subsequences of arbitrarily large discrepancy. |