838 research outputs found
Stolarsky's conjecture and the sum of digits of polynomial values
Let denote the sum of the digits in the -ary expansion of an
integer . In 1978, Stolarsky showed that He conjectured that, as for , this limit
infimum should be 0 for higher powers of . We prove and generalize this
conjecture showing that for any polynomial with and and any base , For any we
give a bound on the minimal such that the ratio . Further, we give lower bounds for the number of such that
.Comment: 13 page
The sum of digits of and
Let denote the sum of the digits in the -ary expansion of an
integer . In 2005, Melfi examined the structure of such that . We extend this study to the more general case of generic and
polynomials , and obtain, in particular, a refinement of Melfi's result.
We also give a more detailed analysis of the special case , looking
at the subsets of where for fixed .Comment: 16 page
Two-dimensional self-affine sets with interior points, and the set of uniqueness
Let be a real matrix with both eigenvalues less than~1 in
modulus. Consider two self-affine contraction maps from , \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M
v + u, \end{equation*} where . We are interested in the properties of
the attractor of the iterated function system (IFS) generated by
and , i.e., the unique non-empty compact set such that . Our two main results are as follows:
1. If both eigenvalues of are between and in
absolute value, and the IFS is non-degenerate, then has non-empty interior.
2. For almost all non-degenerate IFS, the set of points which have a unique
address is of positive Hausdorff dimension -- with the exceptional cases fully
described as well.
This paper continues our work begun in [11].Comment: 29 pages, 7 figure
Multidimensional self-affine sets: non-empty interior and the set of uniqueness
Let be a contracting matrix. In this paper we consider the
self-affine iterated function system , where is a cyclic
vector. Our main result is as follows: if , then the
attractor has non-empty interior.
We also consider the set of points in which have a
unique address. We show that unless belongs to a very special (non-generic)
class, the Hausdorff dimension of is positive. For this special
class the full description of is given as well.
This paper continues our work begun in two previous papers.Comment: 10 pages, no figure
A lower bound for the dimension of Bernoulli convolutions
Let and let denote Garsia's entropy for the
Bernoulli convolution associated with . In the present paper
we show that for all and improve this bound
for certain ranges. Combined with recent results by Hochman and
Breuillard-Varj\'u, this yields for all
. In addition, we show that if an algebraic is such that
for some , then
. Such is, for instance, any root of a Pisot number which is
not a Pisot number itself.Comment: 8 pages, no figure
Open maps: small and large holes with unusual properties
Let be a two-sided subshift on a finite alphabet endowed with a mixing
probability measure which is positive on all cylinders in . We show that
there exist arbitrarily small finite overlapping union of shifted cylinders
which intersect every orbit under the shift map.
We also show that for any proper subshift of there exists a finite
overlapping unions of shifted cylinders such that its survivor set contains
(in particular, it can have entropy arbitrarily close to the entropy of ).
Both results may be seen as somewhat counter-intuitive.
Finally, we apply these results to a certain class of hyperbolic algebraic
automorphisms of a torus.Comment: 15 pages, no figure
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