Search results for "Arithmetic progression"
showing 4 items of 4 documents
The arithmetic decomposition of central Cantor sets
2018
Abstract Every central Cantor set of positive Lebesgue measure is the arithmetic sum of two central Cantor sets of Lebesgue measure zero. Under some mild condition this result can be strengthened by stating that the summands can be chosen to be C s regular if the initial set is of this class.
On a Linear Diophantine Problem of Frobenius: Extending the Basis
1998
LetXk={a1, a2, …, ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1 xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d, …, a+(k−1) d, gcd(a, d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A, c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give …
Resonance between Cantor sets
2007
Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where $\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\dim(K)+\dim(K') \le 1$ (``geometric resonance''), then there exists $r<1$ such that all contraction ratios of the similitudes defining $K$ and $K'$ are powers of $r$ (``algebraic resonance…
A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold
1994
AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.