4. Ries LAG, Miller BA, Hankey BF, Kosary CL, Harras A, Edwards BK, eds. SEER Cancer Statistics Review, 1973-1991: Tables and Graphs. NIH Publ No. 94-2789. Bethesda, MD:National Cancer Institute, 1994. 5. Cantor KP. Drinking water and cancer. Cancer Causes Control 8(3):292-308 (1997). 6. Bates MN, Smith AH, Cantor KP. Case-control study of bladder cancer and arsenic in drinking water. Am J Epid...

We prove the consistency (modulo supercompact) of a negative answer to the Cantor discontinuum partition problem (i.e., some Hausdorff compact space cannot be partitioned to two sets not containing a closed copy of Cantor discontinuum). In this model we have CH. Without CH we get consistency results using a pcf assumption, close relatives of which are necessary for such results; so we try to de...

For a large class of dyadic homogeneous Cantor distributions in R, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to 1 3 . For...

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but Leb(F...

For a Polish group G let covG be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(M) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups covG = cov(M). For example the equality h...

We give an example of Cantor-type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in the Stahl–Totik sense. c ⃝ 2014 Elsevier Inc. All rights reserved. MSC: 30C85; 31A15; 28A78; 28A80

First, we reconstruct Wim Veldman’s result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov’s Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov’s Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift ca...

We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

We construct a minimal topological action Φ̃ of a non-amenable group on a Cantor set C, which is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ̃, C, μ1) and (Φ̃, C, μ2) are not orbit equivalent measurable equivalence relations. AMS Subject Classifications: 37A20

We construct an amenable action ˆ of a non-amenable group on a discrete space. This action extends to a minimal topological action ẑ of on a Cantor set C . We show that ẑ is non-uniquely ergodic and furthermore there exist ergodic invariant measures 1 and 2 such that . ẑ ; C; 1/ and . ẑ ; C; 2/ are not orbit equivalent measurable equivalence relations. This also provides an instance of the fail...