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Publications (10 of 43) Show all publications
Klurman, O. & Kurlberg, P. (2019). A note on multiplicative automatic sequences. Comptes rendus. Mathematique, 357(10), 752-755
Open this publication in new window or tab >>A note on multiplicative automatic sequences
2019 (English)In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 357, no 10, p. 752-755Article in journal (Refereed) Published
Abstract [en]

We prove that any q-automatic completely multiplicative function f: N -> C essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Gold-makher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative functions.

Place, publisher, year, edition, pages
ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-265162 (URN)10.1016/j.crma.2019.10.002 (DOI)000496915600002 ()2-s2.0-85073997485 (Scopus ID)
Note

QC 20191219

Available from: 2019-12-19 Created: 2019-12-19 Last updated: 2019-12-19Bibliographically approved
Holmin, S., Kurlberg, P. & Månsson, D. (2018). On the free path length distribution for linear motion in an n-dimensional box. Journal of Physics A: Mathematical and Theoretical, 51(46), Article ID 465201.
Open this publication in new window or tab >>On the free path length distribution for linear motion in an n-dimensional box
2018 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 46, article id 465201Article in journal (Refereed) Published
Abstract [en]

We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as R -> infinity the free path length coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N -> infinity, and give an explicit (again piecewise real analytic) formula for its probability density function. Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD, 2018
Keywords
free, path, length, distribution
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-246299 (URN)10.1088/1751-8121/aae5ee (DOI)000460029900001 ()2-s2.0-85055489270 (Scopus ID)
Note

QC 20190321

Available from: 2019-03-22 Created: 2019-03-22 Last updated: 2019-05-13Bibliographically approved
Kurlberg, P. & Wigman, I. (2018). Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves. Advances in Mathematics, 330, 516-552
Open this publication in new window or tab >>Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves
2018 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 330, p. 516-552Article in journal (Refereed) Published
Abstract [en]

This is a manuscript containing the full proofs of results announced in [10], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions. (C) 2018 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2018
Keywords
Nodal components, Nazarov-Sodin constant, Spectral measure, Weak-* topology, Continuity, Arithmetic random waves
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-249657 (URN)10.1016/j.aim.2018.03.026 (DOI)000431472100015 ()2-s2.0-85056237337 (Scopus ID)
Note

QC 20190415

Available from: 2019-04-15 Created: 2019-04-15 Last updated: 2019-04-15Bibliographically approved
Kurlberg, P. & Ueberschär, H. (2017). Superscars in the Seba billiard. Journal of the European Mathematical Society (Print)
Open this publication in new window or tab >>Superscars in the Seba billiard
2017 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, To appear in J. Eur. Math. Soc. (JEMS)Article in journal (Refereed) Accepted
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-197970 (URN)
Note

QCR 20170117

Available from: 2016-12-09 Created: 2016-12-09 Last updated: 2017-11-29Bibliographically approved
Amerik, E., Kurlberg, P., Nguyen, K. D., Towsley, A., Viray, B. & Voloch, J. F. (2016). Evidence for the Dynamical Brauer-Manin Criterion. Experimental Mathematics, 25(1), 54-65
Open this publication in new window or tab >>Evidence for the Dynamical Brauer-Manin Criterion
Show others...
2016 (English)In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 25, no 1, p. 54-65Article in journal (Refereed) Published
Abstract [en]

Let phi: X -> X be a morphism of a variety over a number field K. We consider local conditions and a "Brauer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P is an element of X(K) to be disjoint from a subvariety V subset of X, i.e., for V boolean AND O-phi (P) = empty set. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection V boolean AND O-phi (P); this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.

Place, publisher, year, edition, pages
Taylor & Francis, 2016
Keywords
arithmetic dynamics, Brauer-Manin obstruction
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-180209 (URN)10.1080/10586458.2015.1056889 (DOI)000365894600005 ()2-s2.0-84948427877 (Scopus ID)
Note

QC 20160119

Available from: 2016-01-19 Created: 2016-01-08 Last updated: 2017-11-30Bibliographically approved
Kurlberg, P. & Wigman, I. (2016). On probability measures arising from lattice points on circles. Mathematische Annalen, 1-42
Open this publication in new window or tab >>On probability measures arising from lattice points on circles
2016 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, p. 1-42Article in journal (Refereed) Published
Abstract [en]

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice (Formula presented.), gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a “fractal” structure. This complicated structure in some sense arises from prime powers—singularities do not occur for circles of radius (Formula presented.) if n is square free.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186995 (URN)10.1007/s00208-016-1411-4 (DOI)000398175700005 ()2-s2.0-84964290788 (Scopus ID)
Note

QC 20160520

Available from: 2016-05-20 Created: 2016-05-16 Last updated: 2017-04-28Bibliographically approved
Kurlberg, P. & Rosenzweig, L. (2016). Superscars for arithmetic toral point scatterers. Communications in Mathematical Physics, 349(1), 329-360
Open this publication in new window or tab >>Superscars for arithmetic toral point scatterers
2016 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 1, p. 329-360Article in journal (Refereed) Published
Abstract [en]

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori in dimensions . Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that superscars occur-there is phase space localization along families of closed orbits, in the sense that some semiclassical measures contain a finite number of Lagrangian components of the form , for uniformly bounded from below. In particular, for both and , eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues. For , we also show that some semiclassical measures have both strongly localized momentum marginals and non-uniform quantum limits (i.e., the position marginals are non-uniform). For , superscarred eigenstates are quite rare, but for we show that the phenomenon is quite common-with denoting the counting function for the new eigenvalues below x, there are eigenvalues with the property that any semiclassical limit along these eigenvalues exhibits superscarring.

Place, publisher, year, edition, pages
Springer, 2016
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-198016 (URN)10.1007/s00220-016-2749-x (DOI)000392061000007 ()2-s2.0-84992046642 (Scopus ID)
Note

QC 20161213

Available from: 2016-12-09 Created: 2016-12-09 Last updated: 2017-11-29Bibliographically approved
Kurlberg, P. & Wigman, I. (2015). Non-universality of the Nazarov-Sodin constant. Comptes rendus. Mathematique, 353(2), 101-104
Open this publication in new window or tab >>Non-universality of the Nazarov-Sodin constant
2015 (English)In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 353, no 2, p. 101-104Article in journal (Refereed) Published
Abstract [en]

We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-161112 (URN)10.1016/j.crma.2014.09.026 (DOI)000349200000003 ()2-s2.0-84921048169 (Scopus ID)
Funder
Swedish Research Council, 621-2011-5498EU, FP7, Seventh Framework Programme, 335141
Note

QC 20150323

Available from: 2015-03-23 Created: 2015-03-09 Last updated: 2017-12-04Bibliographically approved
Kurlberg, P., Luca, F. & Shparlinski, I. E. (2015). On the fixed points of the map x→xx modulo a prime. Mathematical Research Letters, 22(1), 141-168
Open this publication in new window or tab >>On the fixed points of the map x→xx modulo a prime
2015 (English)In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 22, no 1, p. 141-168Article in journal (Refereed) Published
Abstract [en]

In this paper, we show that for almost all primes p there is an integer solution xε [2,p-1] to the congruence xx ≡ x (mod p). The solutions can be interpretated as fixed points of the map x→xx (mod p), and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.

Keywords
Statistics, Residues, NN
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-166913 (URN)000353050500008 ()2-s2.0-84927591623 (Scopus ID)
Funder
Swedish Research CouncilKnut and Alice Wallenberg Foundation
Note

QC 20150529

Available from: 2015-05-29 Created: 2015-05-21 Last updated: 2017-12-04Bibliographically approved
Freiberg, T. & Kurlberg, P. (2014). On the Average Exponent of Elliptic Curves Modulo p. International mathematics research notices, 2014(8), 2265-2293
Open this publication in new window or tab >>On the Average Exponent of Elliptic Curves Modulo p
2014 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2014, no 8, p. 2265-2293Article in journal (Refereed) Published
Abstract [en]

Given an elliptic curve E defined over <inline-graphic xlink:href="RNS280IM1" xmlns:xlink="http://www.w3.org/1999/xlink"/> and a prime p of good reduction, let <inline-graphic xlink:href="RNS280IM2" xmlns:xlink="http://www.w3.org/1999/xlink"/> denote the group of <inline-graphic xlink:href="RNS280IM3" xmlns:xlink="http://www.w3.org/1999/xlink"/>-points of the reduction of E modulo p, and let e(p) denote the exponent of this group. Assuming a certain form of the generalized Riemann hypothesis (GRH), we study the average of e(p) as <inline-graphic xlink:href="RNS280IM4" xmlns:xlink="http://www.w3.org/1999/xlink"/> ranges over primes of good reduction, and find that the average exponent essentially equals p center dot c(E), where the constant c(E)> 0 depends on E. For E without complex multiplication (CM), c(E) can be written as a rational number (depending on E) times a universal constant, <inline-graphic xlink:href="RNS280IM5" xmlns:xlink="http://www.w3.org/1999/xlink"/>, the product being over all primes q. Without assuming GRH, we can determine the average exponent when E has CM, as well as give an upper bound on the average in the non-CM case.

Keywords
Artin Conjecture, Reductions, Cyclicity, Fields
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-136366 (URN)10.1093/imrn/rns280 (DOI)000334361600008 ()2-s2.0-84896337544 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20140228

Available from: 2013-12-04 Created: 2013-12-04 Last updated: 2017-12-06Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4734-5092

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