Fractional Local Dimension

Fuzzy Local Fractional Differential Equations

Boolean Dimension and Local Dimension

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form “is x < y?”. This application motivates looking for stronger notions of dimensi...

Arithmetic progressions in sets of fractional dimension

Let E ⊂ R be a closed set of Hausdorff dimension α. We prove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions. Mathematics Subject Classification: 28A78, 42A32, 42A38, 42A45, 11B25.

Sparse Potentials with Fractional Hausdorff Dimension

Abstract. We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimensi...

Packing-dimension Profiles and Fractional Brownian Motion

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd (2001) introduced alternate s-dimensional packing dimension profiles P-dims and proved, among many other things, that P-dimsE = DimsE for all integers s > 0 and all analytic sets ...