Remark on estimates of orthomassivity

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Xt(x) = { 1 if 0 ≤ x ≤ t, 0 if t < x ≤ 1, the order relation OMn(K) log n, n → ∞, (4) was noted in [1], while (2) yields only the trivial estimate OMn(K) ≥ 1. The estimate of the quantities (1) is related to other problems in the theory of orthogonal series and combinatorics. So relation (4) actually is equivalent to the combination of the classical Men’shov– Rademacher and Men’shov theorems on convergence and divergences almost everywhere of orthogonal series (see [2, Chap. 9]). Thus, these classical theorems can be regarded as assertions on the complexity of the family of characteristic functions of the intervals (3). In the statement given below which generalizes, in effect, the proof of the Men’shov–Rademacher theorem, we establish an upper bound for orthomassivity; this bound leads to sharp results in a number of cases. In what follows, by#A we denote the number of elements in a finite set A. E-mail: kashin@mi.ras.ru