Recent news concerning the Erdos discrepancy problem

Gowers's Weblog

I’ve just learnt from a reshare by Kevin O’Bryant of a post by Andrew Sutherland on Google Plus that a paper appeared on the arXiv today with an interesting result about the Erd?s discrepancy problem, which was the subject of a Polymath project hosted on this blog four years ago.

The problem is to show that if $latex (epsilon_n)$ is an infinite sequence of $latex pm 1$s, then for every $latex C$ there exist $latex d$ and $latex m$ such that $latex sum_{i=1}^mepsilon_{id}$ has modulus at least $latex C$. This result is straightforward to prove by an exhaustive search when $latex C=2$. One thing that the Polymath project did was to discover several sequences of length 1124 such that no sum has modulus greater than 2, and despite some effort nobody managed to find a longer one. That was enough to convince me that 1124 was the correct bound.


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