456 research outputs found
Are lines much bigger than line segments?
We pose the following conjecture:
(*) If A is the union of line segments in R^n, and B is the union of the
corresponding full lines then the Hausdorff dimensions of A and B agree.
We prove that this conjecture would imply that every Besicovitch set (compact
set that contains line segments in every direction) in R^n has Hausdorff
dimension at least n-1 and (upper) Minkowski dimension n. We also prove that
conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in
particular it holds in the plane.Comment: minor corrections, "much" was added in the titl
Is Lebesgue measure the only -finite invariant Borel measure?
R.D.Mauldin asked if every translation invariant -finite Borel
measure on \RR^d is a constant multiple of Lebesgue measure. The aim of this
paper is to show that the answer is "yes and no", since surprisingly the answer
depends on what we mean by Borel measure and by constant. We present Mauldin's
proof of what he called a folklore result, stating that if the measure is only
defined for Borel sets then the answer is affirmative. Then we show that if the
measure is defined on a -algebra \emph{containing} the Borel sets then
the answer is negative. However, if we allow the multiplicative constant to be
infinity, then the answer is affirmative in this case as well. Moreover, our
construction also shows that an isometry invariant -finite Borel
measure (in the wider sense) on \RR^d can be non--finite when we
restrict it to the Borel sets
On the determination of sets by their triple correlation in finite cyclic groups
Let be a finite abelian group and a subset of it. Suppose that we
know for all subsets of of size up to for how many the
translate is contained in . This information is collectively called
the -deck of . One can naturally extend the domain of definition of the
-deck to include functions on . Given the group when is the -deck
of a set in sufficient to determine the set up to translation? The 2-deck
is not sufficient (even when we allow for reflection of the set, which does not
change the 2-deck) and the first interesting case is . We further restrict
to be cyclic and determine the values of for which the 3-deck of a
subset of \ZZ_n is sufficient to determine the set up to translation. This
completes the work begun by Gr\"unbaum and Moore as far as the 3-deck is
concerned. We additionally estimate from above the probability that for a
random subset of \ZZ_n there exists another subset, not a translate of the
first, with the same 3-deck. We give an exponentially small upper bound when
the previously known one was .Comment: 17 page
The Fixed Point of the Composition of Derivatives
We give an affirmative answer to a question of K. Ciesielski by showing that
the composition of two derivatives always has a
fixed point. Using Maximoff's Theorem we obtain that the composition of two
Darboux Baire-1 functions must also have a fixed point
Reconstructing geometric objects from the measures of their intersections with test sets
Let us say that an element of a given family \A of subsets of can be
reconstructed using test sets if there exist
such that whenever A,B\in \A and the Lebesgue measures of and agree for each then . Our goal will be to find the
least such .
We prove that if \A consists of the translates of a fixed reasonably nice
subset of then this minimum is . In order to obtain this result we
reconstruct a translate of a fixed function using test sets as well, and
also prove that under rather mild conditions the measure function f_{K,\theta}
(r) = \la^{d-1} (K \cap \{x \in \RR^d : = r\}) of the sections of
is absolutely continuous for almost every direction . These proofs
are based on techniques of harmonic analysis.
We also show that if \A consists of the magnified copies of a fixed reasonably nice set , where , then
test sets reconstruct an element of \A. This fails in : we prove
that an interval, and even an interval of length at least 1 cannot be
reconstructed using 2 test sets.
Finally, using randomly constructed test sets, we prove that an element of a
reasonably nice -dimensional family of geometric objects can be
reconstructed using test sets. A example from algebraic topology shows
that is sharp in general
Self-similar and self-affine sets; measure of the intersection of two copies
Let K be a self-similar or self-affine set in R^d, let \mu be a self-similar
or self-affine measure on it, and let G be the group of affine maps,
similitudes, isometries or translations of R^d. Under various assumptions (such
as separation conditions or we assume that the transformations are small
perturbations or that K is a so called Sierpinski sponge) we prove theorems of
the following types, which are closely related to each other;
Non-stability: There exists a constant c<1 such that for every g\in G we have
either \mu(K\cap g(K)) <c \mu(K) or K\subset g(K).
Measure and topology: For every g\in G we have \mu(K\cap g(K)) > 0 \iff int_K
(K\cap g(K)) is nonempty (where int_K is interior relative to K).
Extension: The measure \mu has a G-invariant extension to R^d.
Moreover, in many situations we characterize those g's for which \mu(K\cap
g(K) > 0, and we also get results about those 's for which g(K)\su K or
holds
Squares and their centers
We study the relationship between the sizes of two sets when contains either the whole boundary, or the four
vertices, of a square with axes-parallel sides and center in every point of
, where size refers to one of cardinality, Hausdorff dimension, packing
dimension, or upper or lower box dimension. Perhaps surprinsingly, the results
vary depending on the notion of size under consideration. For example, we
construct a compact set of Hausdorff dimension which contains the
boundary of an axes-parallel square with center in every point , but
prove that such a must have packing and lower box dimension at least
, and show by example that this is sharp. For more general sets
of centers, the answers for packing and box counting dimensions also differ.
These problems are inspired by the analogous problems for circles that were
investigated by Bourgain, Marstrand and Wolff, among others.Comment: 20 pages, no figure
Invariant decomposition of functions with respect to commuting invertible transformations
Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real
functions f that decompose into the sum of a_j-periodic functions, i.e.,
f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a
decomposition if and only if for all partitions to B_1, B_2,... B_N of
{a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least
common multiples b_j, one has D_{b_1}... D_{b_N}f=0.
Actually, we prove a more general result for periodic decompositions of real
functions f defined on an Abelian group A, and, in fact, we even consider
invariant decompositions of functions f defined on some abstract set A, with
respect to commuting, invertible self-mappings of the set A.
We also extend our results to functions between torsion free Abelian groups.
As a corollary we also obtain that on a torsion free Abelian group the
existence of a real valued periodic decomposition of an integer valued function
implies the existence of an integer valued periodic decomposition with the same
periods
Better bounds for planar sets avoiding unit distances
A -avoiding set is a subset of that does not contain pairs
of points at distance . Let denote the maximum fraction
of that can be covered by a measurable -avoiding set. We
prove two results. First, we show that any -avoiding set in
() that displays block structure (i.e., is made up of blocks such that
the distance between any two points from the same block is less than and
points from distinct blocks lie farther than unit of distance apart from
each other) has density strictly less than . For the special case of
sets with block structure this proves a conjecture of Erd\H{o}s asserting that
. Second, we use linear programming and harmonic
analysis to show that .Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to
verify the application of Theorem 3.3. Download the article source to get the
scrip
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