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Arnold W. Miller

Publications and source records attributed to Arnold W. Miller.

At least 37 records · Page 2Linked to original sources
arXiv2004

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group G let cov_G 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(meager) 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 cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. Most recent version at: www.math.wisc.edu/~miller

math.LO↗
arXiv2004

Half of an inseparable pair

A classical theorem of Luzin is that the separation principle holds for the Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B which is disjoint from A but cannot be separated from A by a Delta^0_alpha set C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper Pi^1_1 set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell. Latest version at: www.math.wisc.edu/~miller/

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arXiv2004

The cardinal characteristic for relative gamma-sets

For $X$ a separable metric space define $\pp(X)$ to be the smallest cardinality of a subset $Z$ of $X$ which is not a relative $\ga$-set in $X$, i.e., there exists an $\om$-cover of $X$ with no $\ga$-subcover of $Z$. We give a characterization of $\pp(2^\om)$ and $\pp(\om^\om)$ in terms of definable free filters on $\om$ which is related to the psuedointersection number $\pp$. We show that for every uncountable standard analytic space $X$ that either $\pp(X)=\pp(2^\om)$ or $\pp(X)=\pp(\om^\om)$. We show that both of following statements are each relatively consistent with ZFC: (a) $\pp=\pp(\om^\om) < \pp(2^\om)$ and (b) $\pp < \pp(\om^\om) =\pp(2^\om)$

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arXiv2004

On squares of spaces and Fsigma-sets

We show that the continuum hypothesis implies there exists a Lindelof space X such that X x X is the union of two metrizable subspaces but X is not metrizable. This gives a consistent solution to a problem of Balogh, Gruenhage, and Tkachuk. The main lemma is that assuming the continuum hypothesis there exist disjoint sets of reals X and Y such that X is Borel concentrated on Y, (i.e., for any Borel set B if Y is contained in B then X-B is countable,) but (X x X - diagonal) is relatively Fsigma in (X x X) U (Y x Y).

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arXiv2003

The gamma - Borel conjecture

In this paper we show that it is relatively consistent with ZFC that every gamma-set is countable while not every strong measure zero set is countable. This answers a question of Paul Szeptycki. A set is a gamma-set iff every omega-cover contains a gamma-subcover. An open cover is an omega-cover iff every finite set is covered by some element of the cover. An open cover is a gamma-cover iff every element of the space is in all but finitely many elements of the cover. Gerlits and Nagy proved that every gamma-set has strong measure zero. We also show that is consistent that every strong gamma-set is countable while there exists an uncountable gamma-set. On the other hand every strong measure zero set is countable iff every set with the Rothberger property is countable.

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arXiv2003

Ultrafilters with property (s)

A set X which is a subset of the Cantor set has property (s) (Marczewski (Spzilrajn)) iff for every perfect set P there exists a perfect set Q contained in P such that Q is a subset of X or Q is disjoint from X. Suppose U is a nonprincipal ultrafilter on omega. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA for ctble posets) there exists an ultrafilter U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of omega. http://www.math.wisc.edu/~miller/res/index.html miller@math.wisc.edu

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arXiv2003

On relatively analytic and Borel subsets

Define z to be the smallest cardinality of a function f:X->Y with X and Y sets of reals such that there is no Borel function g extending f. In this paper we prove that it is relatively consistent with ZFC to have b<z where b is, as usual, smallest cardinality of an unbounded family in w^w. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists a set of reals X such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.

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arXiv2003

A nonhereditary Borel-cover gamma-set

In this paper we prove that if there is a Borel-cover gamma-set of cardinality the continuum, then there is one which is not hereditary. A set of reals X is a Borel-cover gamma-set iff for every countable family of Borel sets which is an omega-cover contains a gamma-cover. This is also denoted S_1(Borel_omega, Borel_gamma). This result partially answers a question of Bartoszynski and Tsaban. Tsaban points out that it also gives an example of set which is both a gamma-set and sigma-set but is not hereditarily gamma, which answers a question of Bukovsky, Reclaw, and Repicky.

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arXiv2002

A Mad Q-set

A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $å$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G-delta set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set.

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arXiv2002

On lambda'-sets

A subset X of the Cantor space, 2^ω, is a lambda-prime-set iff for every countable subset Y of the Cantor space Y is relatively G-delta in X union Y. In this paper we prove two forcing results about lambda-prime-sets. First we show that it is consistent that every lambda-prime-set is a gamma-set. Secondly we show that is independent whether or not every dagger-lambda-set is a lambda-prime-set.

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arXiv1996

Introduction to mathematical logic - A problem solving course

This is a set of 288 questions written for a Moore-style course in Mathematical Logic. I have used these (or some variation) four times in a beginning graduate course. Topics covered are: propositional logic axioms of ZFC wellorderings and equivalents of AC ordinal and cardinal arithmetic first order logic, and the compactness theorem Lowenheim-Skolem theorems Turing machines, Church's Thesis completeness theorem and first incompleteness theorem undecidable theories second incompleteness theorem

math.LO↗
arXiv1995

Orthogonal Families of Real Sequences

For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real sequences x such that |x(0)|^p + |x(1)|^p + .. converges. For Hilbert space, l_2, any family of pairwise orthogonal sequences must be countable. Thm 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of l_p for every p>2. It was already known that there exists a family of continuum many pairwise orthogonal elements of real sequences. Thm 2. There exists a perfect maximal orthogonal family of elements of real sequences. Abian raised the question of what are the possible cardinalities of maximal orthogonal families. Thm 3. In the Cohen real model there is a maximal orthogonal set cardinality omega_1, but there is no maximal orthogonal set of cardinality k with ω_1< k < c. Thm 4. For any countable standard model M of ZFC and cardinal k in M such that M satisfies k^ω=k, there exists a ccc generic extension M[G] such that the continuum of M[G] is k and in M[G] for every infinite cardinal i less than or equal to k there is a maximal orthogonal family of cardinality i. Thm 5. (MA_k(σ-centered)) Suppose cardinality of X is less than or equal to k, X contains only finitely many elements of l_2, and for every distinct pair x,y in X the inner product (x,y) converges. Then there exists a z such that z is orthogonal to every element of X. Thm 6.(a) There exists X which is a maximal orthogonal family in l_2 such that for all n with 1\leq n \leqωthere exists Y of cardinality n with (X union Y) a maximal orthogonal family in $\rr^\om$. Furthermore, every maximal orthogonal family containing $X$ is countable. (b) There exists a perfect maximal orthogonal family P such that (P intersect l_2) is a maximal orthogonal family in $l_2$.

math.LO↗
arXiv1995

The combinatorics of open covers (II)

We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Hurewicz property in one case and in the Menger property in other). We consider for each property the smallest cardinality of metric space which fails to have that property. In each case this cardinal turns out to equal another well-known cardinal less than the continuum. We also disprove (in ZFC) a conjecture of Hurewicz which is analogous to the Borel conjecture. Finally, we answer several questions from Part I concerning partition properties of covers.

math.LO↗
arXiv1994

Measurability of functions with approximately continuous vertical sections and measurable horizontal sections

A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that approximately continuous functions are Baire 1., i.e., pointwise For any f:R^2 -> R define f_x(y) = f^y(x) = f(x,y). A function f:R^2 -> R is separately continuous if f_x and f^y are continuous for every x,y in R. Lebesgue in his first paper proved that any separately continuous function is Baire 1. Sierpinski showed that there exists a nonmeasurable f:R^2 -> R which is separately Baire 1. In this paper we prove: Thm 1. Let f:R^2 -> R be such that f_x is approximately continuous and f^y is Baire 1 for every x,y in R. Then f is Baire 2. Thm 2. Suppose there exists a real-valued measurable cardinal. Then for any function f:R^2 -> R and countable ordinal i, if f_x is approximately continuous and f^y is Baire i for every x,y in R, then f is Baire i+1 as a function of two variables. Thm 3. (i) Suppose that R can be covered by omega_1 closed null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 2 for every x,y in R. (ii) Suppose that R can be covered by omega_1 null sets. Then there exists a nonmeasurable function f:R^2 -> R such that f_x is approximately continuous and f^y is Baire 3 for every x,y in R. Thm 4. In the random real model for any function f:R^2 -> R if f_x is approximately continuous and f^y is measurable for every x,y in R, then f is measurable as a function of two variables.

math.LO↗
arXiv1994

Souslin's Hypothesis and Convergence in Category

A sequence of functions f_n: X -> R from a Baire space X to the reals is said to converge in category iff every subsequence has a subsequence which converges on all but a meager set. We show that if there exists a Souslin Tree then there exists a nonatomic Baire space X such that every sequence which converge in category converges everywhere on a comeager set. This answers a question of Wagner and Wilczynski, Convergence of sequences of measurable functions, Acta Math Acad Sci Hung 36(1980), 125-128.

math.LO↗
arXiv1994

Cardinal invariants concerning functions whose sum is almost continuous

A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function. Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more could be said. We show that the cofinality of AA is greater than the continuum, c. Moreover, we show that it is pretty much all that can be said about AA in ZFC, by showing that AA can be equal to any regular cardinal between c^+ and 2^c (with 2^c arbitrarily large). We also show that AA = AD where AD is defined similarly to AA but for the class of Darboux functions. This solves another problem of Maliszewski and Natkaniec.

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