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For a set $P_N\subset[0,1]^2$ of $N\ge2$ points, put

$$D_N(u,v)=\#\bigl(P_N\cap([0,u)\times[0,v))\bigr)-Nuv.$$

The source states Halász's bound

$$\int_0^1\int_0^1 |D_N(u,v)|\,du\,dv\ge c\sqrt{\log N}$$

with an absolute $c>0$. It uses counting discrepancy, without division by $N$. The definition is (1.4), p. 145; the bound is (1.14), p. 148; the complex Riesz product proof outline is (4.15)--(4.17), p. 169. Definitions and statement are claims checked; the full argument has not been reverified.

Bears on. Problem 1221: the localized planar sets in Lemma 7.2 have distinct time coordinates. Changing strict to non-strict rectangle boundaries changes the integrand only on finitely many lines, preserving the integral. The source interface therefore matches this imported inequality; the remaining consecutive-gap argument retains its own standing.