Roth’s estimate of the discrepancy of integer sequences is nearly sharp

Abstract

Letg be a coloring of the set {1, ...,N} = [1,N] in red and blue. For each arithmetic progressionA in [1,N], consider the absolute value of the difference of the numbers of red and of blue members ofA. LetR(g) be the maximum of this number over all arithmetic progression (thediscrepancy ofg). Set$R(N) = \mathop {\min }\limits_g R(g)$ over all two-coloringsg. A remarkable result of K. F. Roth gives*R(N)≫N ^1/4. On the other hand, Roth observed thatR(N)≪N ^1/3+ɛ and suggested that this bound was nearly sharp. A. Sárközy disproved this by provingR(N)≪N ^1/3+ɛ. We prove thatR(N)=N ^1/4+o(1) thus showing that Roth’s original lower bound was essentially best possible.

Our result is more general. We introduce the notion ofdiscrepancy of hypergraphs and derive an upper bound from which the above result follows.