2015/09/17 by Terence Tao · 1 voice · 5 citations
#math.CO #math.NT
We show that for any sequence f: \bf N → \-1,+1\ taking values in \-1,+1\, the discrepancy sup_n,d ∈ \bf N |∑j=1n f(jd)| of f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function \bf g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when \bf g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.