# Colloquium

The Mathematical Sciences have a common colloquium which meets about once every three weeks with presentations of some current research topics by department faculty or visitors. Talks will assume only general mathematical background and all faculty and students are invited to participate

The Colloquium will normally take place on Tuesdays from 16:15 to 17:00 in room Ma335 (other meeting times are also possible, see the exact schedule below).

#### Instructions for contributors

Please bear in mind that the audience will include graduate students and researchers of quite different areas than your own. Therefore it would be greatly appreciated if the following aspects would be taken into account:

• Include a comprehensive account of the setting of the problems, where they come from and what the motivation for studying them is.
• Do not assume that the audience know all the definitions. Take sufficient time to allow the audience to grasp the stated definitions.
• It is better if everyone understands the main theorem than if 5 % understand the whole proof.

The saying has it that if you plan a talk for graduate students, then a few professors might understand it. Therefore we suggest that you plan your talk so that even undergraduates will understand it, and then maybe most of the audience will actually do so.

#### 2016-2017 Schedule

DayTimeSpeakerTitleAbstract
5.714:15

Piotr M. Hajac (IMPAN)

From the noncontractibility of compact quantum groups to a noncommutative Borsuk-Ulam-type conjecture

The only contractible compact Hausdorff topological group G is the trivial one. This classical fact is easily equivalent to the statement that there exists a continuous equivariant map from the join G*G to G if and only if G is trivial. Remembering that all continuous equivariant maps from a topological group to itself (shift maps) are homeomorphisms, one can see the above equivalence as a special case of the equivalence of a Borsuk-Ulam-type conjecture for free continuous actions of G on a compact Hausdorff space X, and the homotopic nontriviality of equivariant continuous maps from X to X. The aim of this talk is to explain how the above claims of classical topology, with applications ranging from the Brouwer fixed-point theorem to the Hilbert-Smith conjecture, generalize to the realm of compact quantum groups acting freely on unital C*-algebras. In particular, after translating the Borsuk-Ulam-type conjecture into C*-algebras and extending it to the noncommutative setting, we will prove the noncommutative Borsuk-Ulam-type conjecture for compact quantum groups containing nontrivial finite classical subgroups (torsion), and show some variations of this theorem obtainable through K-theory. Time permitting, we will also discuss very recent results of S.L. Woronowicz and A. Chirvasitu who respectively proved the invertibility of shift maps for arbitrary locally compact quantum groups and classified them in the reduced and full cases. Based on joint work with L. Dabrowski and S. Neshveyev.

24.514:15-15:00Örjan Stenflo (Uppsala University)V-variable image compression

Consider a rectangular image (or picture), and divide, for each fixed positive integer n, the image into 4^n non-overlapping “image pieces” of side lengths (1/2)^n times the side-lengths of the image. Let V be a fixed positive integer. We say that the image is V-variable if the image contains at most V distinct image pieces for any n. We describe a simple algorithm for lossy image compression where a given target image is approximated by a V-variable image. Joint work with Franklin Mendivil, Acadia University, Canada.

18.512:15-15:00VariousResearchplan seminar presentations
17.514:15Zuzana Vaclavikova (University of Ostrava)

Some remarks on permutations which preserves the weighted density

21.411:15Michael Kinyon (University of Denver)Automorphic loops -  a survey and recent progressA loop is automorphic if every inner mapping is an automorphism. Thus automorphic loops include groups as a special case. My talk will start with a survey of automorphic loops, including a potted history and ending with the current state of art. The main open problem is the existence or nonexistence of a finite, simple and nonassociative automorphic loop.
11.416:15Ilmari NieminenCompactness of weighted composition operatorsIn this talk I will look at the problem of characterizing the compactness of weighted composition operators between Banach spaces of analytic functions. I will focus on operators mapping to a weighted supremum-norm type space and give an overview of approaching this problem through estimating the so called essential norm of the operator. I will try to introduce the basic concepts and give a rough outline of recent methods used to solve this problem in a very general setting while avoiding the more technical details involved.
1.316:15Emma LeppäläFrom loops to groups and backIn this talk I shall present a rough introduction to non-associativity and loop theory. We'll look at some key properties of loops through examples and compare the situation to that in group theory. I'll explain how groups can be used to study loops and present you some of our research questions and the advances made towards solving them.
I'll try to avoid any technicalities.
11.116:15Eino Rossi (Universidad Torcuato di Tella, Buenos Aires)On conformal dimensions of self-affine carpets.

We study the dimensions of quasisymmetric images of horizontal self-affine sets. Our method is based on a general study of tangents of mappings and metric spaces. The main results are that the class of quasisymmetric mappings between horizontal carpets is rather small and that horizontal carpets are minimal for Conformal Assouad dimension.

14.1214:15Biswarup DasLocally compact quantum groups and their representations

Pontryagin duality theory of locally compact abelian groups gives a perfect symmetric situation in the category of locally compact abelian groups: Consider a locally compact abelian group G. Let $\hat{G}$ denotes the set of all characters (i.e. continuous group homomorphisms from G to the circle group). Then $\hat{G}$ can be made into a locally compact abelian group and we can recover G from $\hat{G}$ in the sense that $\hat{ \hat{G} }$ is isomorphic to G as topological groups.  $\hat{G}$ is referred to as the "dual group" of G.  This symmetric situation is no longer true if we are dealing with category of non abelian-locally compact groups.

In the 70's and early 80's, two groups of mathematicians:  Kac & Vainerman and  Enock & Schwartz independently  discovered the notion of "Kac-algebras" which can be thought of as generalization of the notion of "locally compact groups" in the sense that locally compact groups are special cases of Kac algebras. They further made the remarkable discovery: instead of category of locally compact groups, if we consider a category of Kac algebras, then a similar symmetry as in "Pontryagin duality for abelian groups" prevails in the sense that "dual" (in a suitable sense) of a Kac algebra K, denoted $\hat{K}$ is again a Kac algebra and moreover $\hat{ \hat{K} }$ is isomorphic to K (the symmetry which featured in Pontryagin duality). To put it differently: "Dual" of a locally compact (non-abelian) group is not necessarily a locally compact group, but if we think of a locally compact group as a Kac algebra, then the "dual" is again a Kac algebra. In the later years, through the works of S.L. Woronowicz, A. Van Daele, J. Kustermans and S. Vaes, the notion of Kac algebras further evolved into something called "quantum groups".

In this talk, first we will give a gentle introduction to the subject of locally compact quantum group, without going into any technicalities, including precisely defining what is meant by "dual" for us. Then we will address a problem which featured in the representation theory of locally compact quantum groups.

8.1116:15Meng WuOn the multiplications by 2 and by 3

For a natural number m, let T_m denote multiplication by m mod 1 on the unit interval [0,1]. In the later 60's, H. Furstenberg made several far-reaching conjectures concerning the "independence" between the dynamics T_2 and T_ 3. In this talk we will present two of these conjectures.

The first one concerns expansions of irrational numbers in base 2 and base 3, it predicts that the T_2 orbit closure and T_3 orbit closure of an irrational number cannot both be small:  the sum of their Hausdorff dimensions is always larger than 1.

A closely related conjecture concerns intersections of T_2 and T_ 3-invariant sets, which asserts that whenever E is a T_2-invariant closed set and F is a T_3-invariant closed set, then for all real numbers u and t, the Hausdorff dimension (dim) of the intersection of E and uF+t is bounded by dim(E)+dim(F)-1 or 0, whichever is larger.

In this talk, we will present our recent solution to the second conjecture. As an application of our result, we also show that the first conjecture is "almost true": it holds outside a set of Hausdorff dimension 0.

We will present a gentle introduction to the background of these conjectures and theirs relationships.

There will be no technical details in the talk.

25.1016:15Teemu TyniInverse scattering for operators of order four on the line

In scattering theory one considers differential operators with special type of scattering solutions; solutions that look like a sum of in-coming and out-going waves. We consider an inverse scattering problem of recovering the unknown coefficients of a non-linear operator of order four on the line. The unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their non-linearity, but they can be discontinuous or singular in their space variable. It turns out that the inverse Born approximation can be used to recover some essential information about the coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients. We discuss the scattering theory and the related theorems and also present numerical examples regarding different types of coefficients.

27.916:15Jukka KemppainenThe H-distributions

We introduce a class of non-negative probability distributions, which has a nice algebraic structure with respect to multiplication, division and taking powers.

The H-distributions or H-function distributions is a class, which fulfills the above requirements. The H-distributions are non-negative probability distributions whose p.d.f.'s are given by the (Fox) H-functions, which are special functions belonging to the class of higher transcendental functions. Many common non-negative distributions such as the exponential, the gamma, the beta and the chi-square distributions are special cases of H-distributions.
30.814:15

András Máthé (University of Warwick)

Measurable equidecompositions and circle squaring

The Banach-Tarski paradox claims that any two bounded sets A, B with non-empty interiors in a three (or higher) dimensional Euclidean space are equidecomposable: we can divide A into finitely pieces, move them by isometries, and obtain a partition of B.

In the plane, Tarski's circle squaring problem from the 1920s asked whether the square is equidecomposable to a disc (of the same area, as the isometry group is amenable).

In 1990 Laczkovich solved this problem using combinatorial arguments and equidistribution theorems. He showed that for any two sets A and B in the n-dimensional Euclidean space with the same non-zero Lebesgue measure and with boundary of box dimension less than n, A is equidecomposable to B (using translations only).

I will discuss these problems and, in particular, present our recent results with Lukasz Grabowski and Oleg Pikhurko that the pieces can be made measurable in all of the above equidecompositions (provided that A and B are measurable and have equal measures).

23.814:15

Fredrik Ekström (Lund University)

On the Fourier dimension

The Fourier dimension of a Borel set in Euclidean space measures the fastest possible decay rate of the Fourier transform among all finite Borel measures concentrated on the set, and is a lower bound for the Hausdorff dimension of the set. Even though it is called a "dimension", it lacks properties usually associated with dimensions, such as stability under unions and invariance under bi-Lipschitz maps.

This talk will present these and other facts about the Fourier dimension.

### Previous years

2008-2009 Schedule

 29.9 Peter Hästö Function spaces and differential equations 13.1 Iwona Wróbel (Technical U. Warsaw) On the numerical range of companion matrices 20.1 Juri Nesterenko (Moscow State U.) On irrationality of certain numbers 3.11 Esa Läärä Analysis of the distribution of survival times of cancer patients 24.11 Jussi Klemelä Non-parametric function estimation 10.12 Tom Hettmansperger (Pennsylvania State U.) TBA 9.3 Pekka Salmi Fourier transform, duality, quantum groups 16.3 Stephen Walter (McMaster U.) Incorporating patient and physician preferences into the design and analysis of clinical studies 30.3 Hannu Oja (U. Tampere) Multivariate Linear Regression Based on Spatial Signs and Ranks 20.4 Markku Niemenmaa On mathematicians bank accounts and finite groups 27.4 Sigfired Carl (Halle-Wittenberg) Multiple Solutions of Nonlinear Elliptic Problems 11.5 Keijo Väänänen Eräiden päättymättömien tulojen aritmeettisista ominaisuuksista (On arithmetic properties of certain infinite products)

#### 2009-2010 Schedule

 21.9 Esa Järvenpää Visibility 12.1 Petteri Harjulehto (Helsinki) Harnack's inequality for solutions to the p(.)-Laplace equation 19.1 Maarit Järvenpää How to turn a needle 2.11 Pirita Paajanen (Helsinki) p-adic numbers, p-adic integrals and counting conjugacy classes 9.11 Mikael Lindström Bergman and Hardy spaces 23.11 Lasse Holmström Scale space analysis with applications to climate research 8.12 Örjan Stenflo (Uppsala) V-variable fractals 8.2 Jorma Arhippainen On generalization of B^*-algebras 15.2 Jaroslav Hancl (Prague) Expression of the real numbers with the help of infinite series 1.3 Heidi Krzywacki (Helsinki) Formation of teacher identity --- early years as a mathematics teacher 17.5 Lizaveta Ihnatsyeva (Aalto) Tangential boundary behavior of Poisson integrals of regular functions

#### Spring Term 2011 Schedule

 17.1 Tapani Matala-Aho Arithmetic with hypergeometric series 7.2 Kenneth Nordström Convexity of the inverss and Moore-Penrose inverse 7.3 Matti Nuortio Birthday special: Nonlinear potential theory 4.4 Valeriy Serov Fixed energy problem for nonlinear Schrödinger equation

#### 2011-2012 Schedule

 22.8 Abdolrahman Razani (Imam Khomeini Int. U.) Existence of solutions to the Riccati differential equation 12.9 Bing Li Some topics on random covering problem 3.1 Ville Suomala Projections of random fractal measures 7.11 Mikko Sillanpää LASSO and Bayes LASSO 14.11 Simona Samardjiska (NTNU, Trondheim) Permutation polynomials and polynomial quasigroups 28.11 Spyridon Dendridos (Jyväskylä) On uniform bounds for the X-ray transform restricted to polynomial curves 16.1 Tamas Keleti (Eötvös Lorand) Hausdorff dimension of metric spaces and Lipschitz maps onto balls 23.1 Antti Käenmäki (Jyväskylä) Some recent results on self-affine sets 6.2 Ut Van Le The Galerkin method for PDEs 5.3 Juhana Siljander Regularity questions for nonlinear parabolic PDEs 2.4 Peter Hästö Mathematics in Industry, specifically Geometry in Architecture 3.5 Adam Skalski (Polish Academy of Sciences) Quantum symmetry groups 7.5 Raimo Kaasila Opettajaksi opiskelevien matematiikkakuva ja siihen vaikuttaminen

#### 2012-2013 Schedule

 8.8 Pablo Shmerkin (University of Surrey) Packing circles and polygons 8.1 Evgeney Lakshtanov (University of Averio) Interior Transmission Eigenvalue Problem 22.1 Zoltan Buczolich (Budapest) TBA 3.11

#### 2014-2015 Schedule

 27.8. Zoltan Buczolich (Eötvös Lorand University, Budapest) On the gradient problem of C.E. Weil 22.9. Wen Wu Hankel determinants of automatic sequences 3.10. Neil Dobbs (University of Helsinki) How wild is exponential? 27.10. Meng Wu An invitation to Multifractal analysis  - with an emphasis on the study of multiple ergodic averages 17.11 Tuomo Ojala (Jyväskylä) Thin and Fat sets in metric spaces 1.12 Andre Juffer (Biocenter Oulu) Inclusion of charge fluctuations in coarse grained molecular dynamics simulations. 8.12 Valtteri Niemi (University of Turku) Design of security architectures for mobile systems 12.1 Henna Koivusalo (University of York) Pattern complexity of cut and project sets 19.1 Kalle Leppälä Raney's algorithm for linear fractional transformations of simple continued fractions 2.2 Peter Hästö Harmonic analysis in generalized Orlicz spaces 20.2 (Fri!) Eino Rossi (University of Jyväskylä) Local structure of self-affine sets 30.3 Lassi Roininen (SGO) Gaussian Markov random fields in Bayesian statistical inverse problems 8.4 (Wed!) Tuomas Sahlsten (The Hebrew University of Jerusalem) Fourier series of singular functions 20.4 Jukka Kemppainen Fractional Calculus - What Is It?" 27.4 Erno Saukko A short introduction to composition operators 11.5 Georgios Fotopoulos Inverse scattering problems for the nonlinear Schrödinger operator in two dimensions 8.6 De-Jun Fung (The Chinese University of Hong Kong) On the topology of integer polynomials with bounded coefficients

#### 2015-2016 Schedule

DayTimeSpeakerTitleAbstract
14.615:15Pablo Shmerkin (Torcuato di Tella University, Buenos Aires)

Flattening of $L^p$ norms under convolution, and the dimension and densities of Bernoulli convolutions.

I will present an inverse theorem for measures that don't flatten the $L^p$ norm exponentially under convolution. As an application, new results on the dimensions and densities of Bernoulli convolutions, outside of a small set of exceptional parameters, are obtained.

9.614:15

Piotr M. Hajac (IMPAN / University of New Brunswick)

There and back again: from the Borsuk-Ulam theorem to quantum spaces

Assuming that both temperature and pressure are continuous functions, we can conclude that there  are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C*-algebras and Gelfand-Naimark Theorems, and carefully avoiding the Mordor of incomprehensible technicalities, we  shall arrive in the Gondor of compact quantum groups acting freely on unital C*-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. To end with, we will pay tribute to the ancient quantum group SUq(2), and show how the proven non-trivializability of Pflaum's SUq(2)-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. Time permitting, we shall also explain how to extend the non-trivializability result from the Pflaum quantum instanton bundle to an arbitrary finitely iterated equivariant join of SUq(2) with itself. The latter is a quantum sphere with a free SUq(2)-action whose space of orbits defines a quantum quaternionic projective space. (Based on joint work with Paul F. Baum, Ludwik Dabrowski and Tomasz Maszczyk.)

24.514:15

Stéphane Seuret (Université Paris Est)

Random sparse sampling in a Gibbs weighted tree.

Consider a Gibbs measure on the dyadic tree, and the associated « Gibbs » dyadic tree where each finite word $w$ carries the weight $\mu([w])$. The length of $w$ is denoted by $|w|$. Fix a parameter $\eta\in (0,1)$. We perform a sampling of the Gibbs tree, by keeping each value $\mu([w])$ with probability $2^{-|w|\eta}$, otherwise we replace this value by 0. We study the possibility of reconstructing the initial Gibbs tree from the sampled tree, and perform the multifractal analysis of the remaining structure (which can be viewed as a capacity on the dyadic tree). Various phase transitions occur, both in the reconstruction process and in the multifractal spectra.

3.516:15Kay Schweiger (University of Helsinki)Synchronizing Quantum SystemsThe speed convergence of Markov processes is crucial in various branches
of mathematics and its applications. In many cases stochastic couplings
provide an effective method to estimate the limit behaviour.

During the talk we will see how estimates can be achieved via particular
codings. Furthermore, we will review typical challenges of applying a
similar approach to quantum systems. We present a construction for
quantum Markov processes that yields similar results and discuss its
consequences and open problems.
26.416:15Laurent Dufloux (Paris XIII)Hausdorff dimension of limit setsThe boundary of complex hyperbolic n-space can be endowed with two different family of metrics: Gromov metrics or spherical metrics. The Hausdorff dimension of limit sets of discrete groups of complex hyperbolic isometries is well understood when the boundary is endowed with Gromov metrics; this is a special case of some very general results. On the other hand, little is known when the boundary is endowed with the more familiar spherical metric. I will explain how the Ledrappier-Young formula sheds some light on this question, and construct some non-trivial examples of discrete subgroups that allow for exact computation.
19.416:15Sari LasanenElliptic boundary value problems for stochastic partial differential equationsWe will consider how to define Neumann and Robin boundary value problems when the source term is Gaussian white noise, which has irregular realizations.
12.416:15Mikko Sillanpää

Rapid Bayesian inference of genetic parameters in mixed linear models.

Short introduction to animal model which is a certain type of mixed linear model, often used in genetics, is first given.

Then efficient MCMC implementation of multiple random effects (genetic effects) in this context is considered. Also another approach presenting decomposition of prior of multivariate normal distribution is shown as well as an analytic approach that avoids MCMC sampling completely.
5.416:15Changhao Chen

Fractal percolation, porosity, and dimension.

Porosity describes the size of holes in a given set. Fractal percolation (or Mandelbrot percolation) is a classical random fractal sets model. In this talk I will show some results on the porosities in Fractal percolation. Joint work with Tuomo Ojala, Eino Rossi, and Ville Suomala.

15.316:15Pekka SalmiIdempotent probability measures and idempotent states
8.316:15Tapani Matala-AhoOn Siegel's lemma
23.216:15Marko Huhtanen

Factoring matrices into the product of diagonal and circulant matrices.

30.1114:15Olli HyvärinenSpectra of invertible weighted composition operators

Let $\varphi$ be an automorphism of the open unit disc $D$. For such $\varphi$, we investigate the spectra and the essential spectra of invertible weighted composition operators $uC_\varphi$ acting on a wide class of analytic function spaces; this class contains, for example, Hardy spaces $H^p$ and standard weighted Bergman spaces $A^p_\alpha$.

23.1112:15Dimitri TuomelaFacilitating shared reasoning through Reversed Equation Solving -intervention

My objective is to build robust design theory and teacher guide around the intervention of Reversed Equation Solving (RES). It aims to have an impact on classroom norms, learners’ conceptual understanding related to equation solving and teachers’ visions of high-quality mathematics instruction. RES intends to serve as an example that shared reasoning can be facilitated around the relevant mathematical ideas related to equation solving which is often seen by teachers as a rote-learning based topic. Results of the pilot study and the initial setting of the design research will be discussed.

23.1112:15Riikka PalkkiMathematics teachers and errors

My PhD study discusses the role of the teacher when encountering errors in mathematics lessons. The teacher's conceptions on errors and the advice given regarding them are considered. The topic is approached by questionnaires, classroom conversations and interviews. A case study example of four classes of seventh graders analyzing a planted error in a teacher lead conversation is discussed.

9.1114:15-15:00Biswarup DasEberlein compactification of quantum groups.

In 1949 Eberlein initiated the theory of weakly almost periodic functions on abelian groups in order to gain understanding of the Fourier-Stieltjes transforms of measures on the dual group. Following his work, in the recent years, there has been a comprehensive study of the notion of "weakly almost periodic functions" on locally compact groups. The weakly almost periodic functions on a locally compact group G form a commutative C*-algebra wap(G), whose character space Gwap gives the largest "semitopological" semigroup compactification of G. An important C* subalgebra of wap(G) is the Eberlein algebra, whose character space gives the largest "Hilbert space representable" semigroup compactification of G. We would like to seek a generalization of Eberlein compactification within the framework of quantum groups. We will first device a category (which depends on G) whose initial object will be the Eberlein compactification. Then we will replace the classical objects in that category by quantum objects (for example, G will be replaced by a quantum group G) and will prove existence of initial object in this new category, which can be taken as the quantum version of the classical initial object (Eberlein compactification). (Based on joint work with Matthew Daws).

26.1014:15-15:00Abel FarkasHow to treat Hausdorff measure for self-similar sets with overlaps.
We introduce a method that can replace the role of the open set condition in proves allowing us to conclude results in the overlapping case. As an application we show that the Hausdorff measure and the Hausdorff content of self-similar sets agree in the critical dimension. As a consequence of this we deduce that if the Hausdorff dimension $t$ of a self-similar set $K$ is strictly less than one then the following are equivalent:
1) the $t$-dimensional Hausdorff measure of $K$ is positive;
2) the defining IFS of $K$ satisfies the weak separation property
28.914:15-15:00Zitiong LiVarying-coefficient models with application in quantitative geneticsIn statistics, varying-coefficient (VC) models, as an extension of classical linear regression, do not assume a regression parameter to be constant, but allow the paramter to depend on other covariates such as time. We consider recent advances of  the VC models concentrating on analyzing high-dimensional data. The methods can be used in quantitative genetics studies to identify phenotype-genetics relationship.
31.8.14:15-15:00Jonathan Fraser (University of Manchester)The Brownian graph has maximal Fourier dimension
24.8.14:15-15:00Bing Li (South China University of Technology)Some topics on random covering sets over the torusLet $\mathbb{T}^d$ be the $d$-dimensional torus and $g_n$ be a sequence of subsets in $\mathbb{T}^d$. A sequence of random sets $G_n=\xi_n+g_n$ is formed by randomly translating $g_n$, where $\{\xi_n\}$ be a sequence of i.i.d. random variables uniformly distributed on $\mathbb{T}^d$. We are interested in the random covering set $E$ of the points in $\mathbb{T}^d$ covered infinitely many times by $\{G_n\}$. In this talk, we will introduce some results concerning the sizes of the set $E$ in the sense of topology, measure and dimension. Some sufficient and necessary conditions for the Dvoretzkoy covering problem, namely almost surely every point is covered infinitely often will be mentioned.
5.8.11:15-12:00

Pablo Shmerkin (Torcuato di Tella University, Buenos Aires)

Absolute continuity of complex Bernoulli convolutions.

Let $\nu_\lambda$ be the distribution of the random sum $\sum_{n=1}^\infty \pm \lambda^n$, where $\lambda$ is a complex number in the open unit disk. This is a natural generalization of Bernoulli convolutions (which correspond to the case of real $\lambda$) to the complex plane. By adapting a method previously developed for the real case, we show that $\nu_\lambda$ is absolutely continuous for all $\lambda$ of modulus larger than $1/2$, outside of a set of possible exceptions of zero Hausdorff dimension. Previously, it was not even known if the exceptional set had zero Lebesgue measure. This is joint work with Boris Solomyak.

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