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On the small ball inequality in three dimensions. (English) Zbl 1202.42007
Let \(h_{R}\) denote an \(L^{\infty}\)-normalized Haar function associated to a dyadic rectangle \(R\subseteq [0,1]^{d}\) with volume \(|R|\). The authors show for all integers \(n\) and coefficients \(\alpha (R)\) \[ 2^{-n}\sum_{|R|=2^{-n}} |\alpha(R)|\ll n^{1-\eta}\;\Big\|\sum_{|R|=2^{-n}} \alpha(R)h_{R}\Big\|_{\infty} \] with \(0<\eta <\frac{1}{2}.\) This is an important improvement over the trivial estimate by an amount of \(n^{-\eta}\) while the “small ball conjecture” says that the inequality should hold with \(\eta= \frac{1}{2}.\) In dimension \(d=2\) this was resolved by M. Talagrand [Ann. Probab. 22, No. 3, 1331–1354 (1994; Zbl 0835.60031)]. V. N. Temlyakov [J. Complexity 11, No. 2, 293–307 (1995; Zbl 0830.42015), East J. Approx. 1, No. 1, 61–72 (1995; Zbl 0845.41011)] has given a simpler proof using ideas of Roth, Schmidt and Halasz. The “small ball problem” has its origin in asymptotic estimates for random processes. Furthermore, there is an important application in the theory of irregularities of distribution obtaining lower bounds for the discrepancy of point sequences [see M. Drmota and R. F. Tichy, Sequences, discrepancies and applications. Lecture Notes in Mathematics. 1651. Berlin: Springer (1997; Zbl 0877.11043)]. In dimension \(d=3\), J. Beck [Compos. Math. 72, No. 3, 269–339 (1989; Zbl 0691.10041)] has obtained a weaker inequality with a logarithmic factor replacing \(n^{1-\eta}.\) This implies an improvement of Roth’s lower discrepancy bound by a \(\log\log n\) factor (in dimension \(d=3\)) whereas the present inequality implies an improvement by a factor \(\log n\) (in arbitrary dimensions \(d\geq 3).\)

MSC:
42A05 Trigonometric polynomials, inequalities, extremal problems
11K38 Irregularities of distribution, discrepancy
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