Under mild conditions on the weight function K we characterize lacunary series in the so-called QK spaces.

Given an integer matrix M ∈ GLn(R) and a point y ∈ R/Z, consider the set Ẽ(M,y) def = { x ∈ R : y / ∈ {Mkx mod Zn : k ∈ N} } . S.G. Dani showed in 1988 that whenever M is semisimple and y ∈ Q/Z, the set Ẽ(M,y) is winning in the sense of W. Schmidt (a property implying density and full Hausdorff dimension). In this paper we strengthen this result, extending it to arbitrary y ∈ R/Z and M ∈ GLn(R)...

In 1975 Philipp showed that for any increasing sequence (nk) of positive integers satisfying the Hadamard gap condition nk+1/nk > q > 1, k ≥ 1, the discrepancy DN of (nkx) mod 1 satisfies the law of the iterated logarithm 1/4 ≤ lim sup N→∞ NDN(nkx)(N log logN) −1/2 ≤ Cq a.e. Recently, Fukuyama computed the value of the lim sup for sequences of the form nk = θ , θ > 1, and in a preceding paper t...

We obtain lacunary recurrent relations with gaps of length four for the Bernoulli and Euler polynomials and, as a consequence, known new numbers.

The rational self-assembly of mono-lacunary Keggin clusters with 4d and 4f metal salts via a conventional method has yielded two novel polyoxometalate-based 4d-4f heterometallic compounds containing lacunary Keggin anions connected to ten metal atoms: {[Ag{Ag2(H2O)4}{Ln(H2O)6}2H ⊂ {SiW11Ln(H2O)4O39}2]·nH2O (Ln = Ce and n = 7 for 1, Ln = Pr and n = 3 for 2). Their structures were determined by s...