In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but neverpublished, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the ...

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...

Organothiophosphoryl polyoxotungstates R(contains)XW(infinityinfinity)O(contains exists) (/-) , R(contains) P(contains)W(infinity),O(infinity) (/-), R(contains)PW( exists)O(contains) (Delta) (-)(X = P, Si, Ge, B or Ga; R = PhP(S), C(6)H(11)P(S)) have been prepared from lacunary polyoxoanions and PhP(S). The products were characterized by elemental analysis, IR, and NMR spectroscopy. According t...

A subset C of an abelian group G is said to be a minimal additive complement W⊆G if C+W=G and C′+W≠G for any proper C′⊂C. In this paper, we study certain subsets the integers that arise or do not as complements. Following suggestion Kwon, show bounded-below sets with arbitrarily large gaps Moreover, our construction shows such set belongs co-minimal pair, strengthening result Biswas Saha lacuna...

In this article we introduce and study the notions of ( , ) s v r  -lacunary strongly summable, ( , ) s v r  Cesàro strongly summable, ( , ) s v r  statistically convergent and ( , ) s v r  -lacunary statistically convergent sequence of fuzzy numbers. Consequently we construct the spaces   ( , ) F sv r L  ,   1 ( , ) F sv r   ,   ( , ) F sv r S  and   ( , ) F sv r S  respecti...