Polymath5 – The Erdős discrepancy problem – is on its way.
After several discussion threads, polymath5 devoted to Erdos’s discrepency problem is on its way on Gowers’s blog. While a theoretical post with several possible attacks on the problem is planned, there is intensive experimental activity. The picture above shows a sequence of +/- signs length 1124 with discrepency 2: namely on every arithmetic progression of the form d, 2d, 3d, … where all terms are between 1 and 1124 the gap between the number of ’+'s and the number of ’-'s is at most two. The present hopes from these experiments are described in this paragraph:
“ I now think it likely that we will come up with a formula that gives something very close to the 1124 sequence. If so, then I am keeping my fingers very firmly crossed that it will give an infinite sequence with sublogarithmic discrepancy. But we shall see: some computation is still needed before we will know what the formula is. There is also a chance that we will obtain a formula but with parameters that have to be chosen for each prime, and that we will not be able to see how to choose them in general. In that case, we may at least obtain a very efficient algorithm for finding extremely long sequences of low discrepancy.”
This is very nice.