Tangent measures of typical measures

We prove that for a typical Radon measure mu in R^d, every non-zero Radon measure is a tangent measure of mu at mu almost every point. This was already shown by T. O'Neil in his PhD thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.Comment: v3: 20 pages, 4 figures, the main result was already proved by T. O'Neil in his PhD thesis with a different proof, added a reference to it. Peer-reviewed versio

Trigonometric series and self-similar sets

Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $\mu$ on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. The rate of $\widehat{\mu}(\xi) \to 0$ is also shown to be logarithmic if $\log r_i / \log r_j$ is diophantine for some $i \neq j$. The proof is based on quantitative renewal theorems for random walks on $\mathbb{R}$.Comment: 18 pages, v2: improved the main theore

Fourier transform of self-affine measures

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $\Gamma$ generated by the matrices $A_j$, $j \in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $\mu = \sum p_j f_j \mu$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehat{\mu}(\xi) \to 0$ as $|\xi| \to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $\Gamma$ is connected real split Lie group in the Zariski topology, then $\widehat{\mu}(\xi)$ has a power decay at infinity. Hence $\mu$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $\Gamma$ and non-compactness of the image of $\Gamma$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehat{\mu}$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.Comment: v2: 27 pages, updated references. Accepted to Advances in Mat

Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. An

Dimension, entropy, and the local distribution of measures

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local homogeneity, porosity and conical densities of measures.Comment: v2: 23 pages, 6 figures. Updated references. Accepted to J. London Math. So

Tangent measures of non-doubling measures

We construct a non-doubling measure on the real line, all tangent measures of which are equivalent to Lebesgue measure.Comment: 17 pages, 5 figures. v2: Minor corrections throughout, and section six completely rewritten in a more reader-friendly style; Accepted to Math. Proc. Cambridge Philos. So

On Fourier analytic properties of graphs

We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of the fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension zero.Comment: 11 pages, 1 figure; references added and typos corrected in v2; to appear in Int. Math. Res. Not. IMR

Tinnitus – psychiatric comorbidity and treatment using transcranial magnetic stimulation (TMS)

Tinnitus is the perception of sound in the absence of any external noise. It severely impairs the quality of life in 1-2% of people. Tinnitus is frequently associated with depression, anxiety, and insomnia. The exact pathophysiology of tinnitus is still unclear. No curative therapy exists for chronic tinnitus, and treatment focuses on symptomatic relief. Repetitive transcranial magnetic stimulation (rTMS) is a non-invasive neuromodulation technique that is used for treating depression and neuropathic pain. The evidence of its efficacy for chronic tinnitus is still inconclusive, and the optimal treatment protocols are thus still obscure. This thesis aimed to further evaluate the use of rTMS for chronic tinnitus and investigate the psychiatric comorbidity of tinnitus patients. The first (open pilot) study utilized electric field (E-field) navigated rTMS for very severe chronic tinnitus with promising results. In the second (randomized placebo-controlled) study, the effects of 1-Hz E-field rTMS targeted according to the tinnitus pitch to the left auditory cortex were analyzed. Despite the significant improvements in tinnitus, active rTMS was not superior to the placebo, possibly due to large placebo-effect and wide inter-individual variation in treatment efficacy. The third study on parallel groups compared the effects of neuronavigated rTMS to nonnavigated rTMS (based on the 10-20 EEG localization system). Both groups benefitted from the treatment, but the method of coil localization was not a critical factor for treatment outcome. In the fourth study, current and lifetime DSM-IV diagnoses of Axis I (psychiatric disorders) and Axis II (personality disorders) were assessed in tinnitus patients using structured clinical interviews (SCID-I and -II). Tinnitus patients were prone to episodes of major depression, and they often had obsessive-compulsive personality features. Psychiatric disorders in this study seemed to be comorbid or predisposing conditions rather than the consequences of tinnitus.Tinnitus – psykiatrinen sairastavuus ja hoito transkraniaalisella magneettistimulaatiolla (TMS) Tinnituksen ääniaistimus syntyy ilman ulkoista äänilähdettä. Se heikentää vakavasti elämänlaatua 1-2%:lla ihmisistä. Tinnitus yhdistetään usein masennukseen, ahdistukseen ja unettomuuteen. Tinnituksen tarkka syntymekanismi on vielä epäselvä. Pitkäaikaiselle tinnitukselle ei ole parantavaa hoitoa, vaan hoidossa keskitytään oireiden lievittämiseen. Transkraniaalinen magneettistimulaatio sarjapulssein (rTMS) on kajoamaton aivojen toimintaa muokkaava menetelmä, jota käytetään masennuksen ja hermoperäisen kivun hoidossa. Sen teho pitkäaikaiseen tinnitukseen on vielä epävarmaa ja optimaaliset hoitoprotokollat ovat selvittämättä. Tämän väitöskirjan tavoitteena oli arvioida rTMS:n käyttöä pitkäaikaisen tinnituksen hoidossa ja lisäksi tutkia tinnituspotilaiden psykiatrista sairastavuutta. Ensimmäisessä osatyössä (avoin pilotti) käytettiin sähkökenttäohjattua (E-field) navigoivaa rTMS:a pitkäaikaiseen, erittäin vaikeaan tinnitukseen lupaavin tuloksin. Toisessa osatyössä (satunnaistettu lumekontrolloitu) arvioitiin tinnitusäänen korkeuden mukaan vasemmalle kuuloaivokuorelle suunnatun 1- Hz:n sähkökentän mukaan navigoidun rTMS:n vaikutuksia. Vaikka tinnitus helpottui merkittävästi, ei aktiivi-rTMS ollut lumehoitoa parempi, mahdollisesti johtuen suuresta lumevaikutuksesta ja laajasta yksilöiden välisestä vaihtelusta hoidon tehossa. Kolmannessa osatyössä verrattiin rinnakkaisryhmien välillä neuronavigoidun rTMS:n ja sokko rTMS:n (10-20 EEG-systeemiin perustuva paikannus) vaikutuksia. Molemmat ryhmät hyötyivät hoidosta, eikä kelan paikannusmenetelmä ollut ratkaiseva tekijä hoidon lopputuloksen kannalta. Neljännessä osatyössä nykyiset ja elämänaikaiset akselin I (psykiatriset häiriöt) ja akselin II (persoonallisuushäiriöt) DSM-IV diagnoosit määritettiin tinnituspotilailta käyttäen strukturoituja psykiatrisia haastatteluja (SCID-I ja -II). Tinnituspotilaat olivat alttiita vakaville masennusjaksoille ja heillä oli usein vaativan persoonallisuuden piirteitä. Psykiatriset häiriöt vaikuttivat olevan ennemmin samanaikaisia tai altistavia tiloja kuin tinnituksen seurauksena ilmaantuneita häiriöitä