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  • 1. Amerik, Ekaterina
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Nguyen, Khoa D.
    Towsley, Adam
    Viray, Bianca
    Voloch, Jose Felipe
    Evidence for the Dynamical Brauer-Manin Criterion2016In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 25, no 1, p. 54-65Article in journal (Refereed)
    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.

  • 2. Benedetto, R. L.
    et al.
    Ghioca, D.
    Hutz, B.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Scanlon, T.
    Tucker, T. J.
    Periods of rational maps modulo primes2013In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 355, no 2, p. 637-660Article in journal (Refereed)
    Abstract [en]

    Let K be a number field, let be a rational map of degree at least 2, and let . We show that if alpha is not in the forward orbit of beta, then there is a positive proportion of primes of K such that is not in the forward orbit of . Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace alpha by a hypersurface, such as the ramification locus of a morphism .

  • 3. Benedetto, Robert L.
    et al.
    Ghioca, Dragos
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tucker, Thomas J.
    A case of the dynamical Mordell-Lang conjecture2012In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 352, no 1, p. 1-26Article in journal (Refereed)
    Abstract [en]

    We prove a special case of a dynamical analogue of the classical Mordell- Lang conjecture. Specifically, let phi be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of phi on (P(1))(g). If the coefficients of phi are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P(1))(g) has only finite intersection with any curve contained in (P(1))(g). We also show that our result holds for indecomposable polynomials phi with coefficients in C. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over C uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (phi, phi) on A(2).

  • 4. Benedetto, Robert L.
    et al.
    Ghioca, Dragos
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tucker, Thomas J.
    A gap principle for dynamics2010In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 146, no 4, p. 1056-1072Article in journal (Refereed)
    Abstract [en]

    Let f(1), ... , f(g) is an element of C(z) be rational functions, let Phi = (f(1), ... ,f(g)) denote their coordinate-wise action on (P-1)(g), let V subset of (P-1)(g) be a proper subvariety, and let P be a point in (P-1)(g)(C). We show that if S = {n >= 0 : Phi(n)(P) is an element of V(C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n <= N such that Phi(n)(P) is an element of V(C) is less than log(k)N, where log(k) denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport-koth and Mumford.

  • 5. Brzezinski, J.
    et al.
    Holsztynski, W.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the congruence ax plus by 1 modulo xy2005In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 14, no 4, p. 391-401Article in journal (Refereed)
    Abstract [en]

    We give bounds on the number of solutions to the Diophantine equation (X + 1/x)(Y + 1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax + by equivalent to 1 modulo xy.

  • 6. Bump, D.
    et al.
    Choi, K. K.
    Kurlberg, Pär
    Vaaler, J.
    A local Riemann hypothesis, I2000In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 233, no 1, p. 1-19Article in journal (Refereed)
  • 7. Fainsilber, L.
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wennberg, B.
    Lattice points on circles and discrete velocity models for the Boltzmann equation2006In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 37, no 6, p. 1903-1922Article in journal (Refereed)
    Abstract [en]

    The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].

  • 8. Fainsilber, Laura
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wennberg, Bernt
    The Boltzmann equation on a two-dimensional lattice; theoretical and numerical results2007In: Bulletin of the Institute of Mathematics, Academia Sinica, ISSN 0304-9825, Vol. 2, no 2, p. 667-685Article in journal (Refereed)
    Abstract [en]

    The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. This paper contains a brief description of the proof that was recently presented elsewhere ([L. Fainsilber, P. Kurlberg, B. Wennberg, preprint 2004]). It also presents the results of numerical experiments.

  • 9.
    Felix, Adam Tyler
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the fixed points of the map x↦xx modulo a prime, II2017In: Finite Fields and Their Applications, ISSN 1071-5797, E-ISSN 1090-2465, Vol. 48, p. 141-159Article in journal (Refereed)
    Abstract [en]

    We study number theoretic properties of the map x↦xx(modp), where x∈{1,2,…,p−1}, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p&lt;N for which the map only has the trivial fixed point x=1. A key technical result, possibly of independent interest, is the existence of subsets Nq⊂{2,3,…,q−1} such that almost all k-tuples of distinct integers n1,n2,…,nk∈Nq are multiplicatively independent (if k is not too large), and |Nq|=q⋅(1+o(1)) as q→∞. For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of Fq-linear forms {Ln}n=2 q−1 “behaves randomly” in the sense that |{v∈Fq d:Ln(v)=1,n=2,3,…,q−1}|∼qd(1−1/q)q∼qd/e. (Here d=π(q−1) and the coefficients of Ln are given by the exponents in the prime power factorisation of n.)

  • 10.
    Freiberg, Tristan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Average Exponent of Elliptic Curves Modulo p2014In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2014, no 8, p. 2265-2293Article in journal (Refereed)
    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.

  • 11. Freiberg, Tristan
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rosenzweig, Lior
    Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers2017In: Communications in Number Theory and Physics, ISSN 1931-4523, E-ISSN 1931-4531, Vol. 11, no 4, p. 837-877Article in journal (Refereed)
    Abstract [en]

    We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime k-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rud-nick and Ueberschar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.

  • 12. Friedlander, John B.
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shparlinski, Igor E.
    PRODUCTS IN RESIDUE CLASSES2008In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 15, no 5-6, p. 1133-1147Article in journal (Refereed)
    Abstract [en]

    We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results "on average" over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.

  • 13. Granville, Andrew
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Poisson statistics via the Chinese Remainder Theorem2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 218, no 6, p. 2013-2042Article in journal (Refereed)
    Abstract [en]

    We consider the distribution of spacings between consecutive elements in subsets of Z/qZ, where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z/q(1)q(2)Z that are created via the Chinese Remainder Theorem from subsets of Z/q(1)Z and Z/q(2)Z (for q(1), q(2) coprime), and give criteria for when the spacings modulo q(1)q(2) are Poisson. Moreover, we also give some examples when the spacings modulo q(1)q(2) are not Poisson, even though the spacings modulo q(1) and modulo q(2) are both Poisson.

  • 14.
    Holmin, Samuel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Jones, Nathan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    McLeman, Cam
    Petersen, Kathleen
    Missing class groups and class number statistics for imaginary quadratic fieldsManuscript (preprint) (Other academic)
    Abstract [en]

    The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h increases without bound, and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance the elementary abelian group of order 27 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F(h) for odd h up to 10^6 and F(G) for G a p-group of odd order with |G| up to 10^6. The numerical evidence matches quite well with our conjectures.

  • 15.
    Holmin, Samuel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Månsson, Daniel
    On the free path length distribution for linear motion in an n-dimensional boxManuscript (preprint) (Other academic)
    Abstract [en]

    We consider the distribution of free path lengths, or the distance betweenconsecutive bounces of random particles, in an n-dimensional rectangular box.If each particle travels a distance R, then, as R → ∞ the free path lengthscoincides with the distribution of the length of the intersection of a randomline with the box (for a natural ensemble of random lines) and we determinethe mean value of the path lengths. Moreover, we give an explicit formula(piecewise real analytic) for the probability density function in dimension twoand three.In dimension two we also consider a closely related model where eachparticle is allowed to bounce N times, as N → ∞, and give an explicit (againpiecewise real analytic) formula for its probability density function.Further, in both models we can recover the side lengths of the box fromthe location of the discontinuities of the probability density functions.

  • 16.
    Holmin, Samuel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Månsson, Daniel
    KTH, School of Electrical Engineering and Computer Science (EECS), Electromagnetic Engineering.
    On the free path length distribution for linear motion in an n-dimensional box2018In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 51, no 46, article id 465201Article in journal (Refereed)
    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.

  • 17. Juul, Jamie
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Madhu, Kalyani
    Tucker, Tom J.
    Wreath Products and Proportions of Periodic Points2016In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 13, p. 3944-3969Article in journal (Refereed)
    Abstract [en]

    Let phi : P-1 -> P-1 be a rational map of degree greater than 1 defined over a number field k with ring of integers o(k). For each prime p of good reduction for phi, we let phi(p) denote the reduction of phi modulo p. A random map heuristic suggests that for large p, the proportion of periodic points of phi(p) in P-1 (o(k)/p) should be small. We show that this is indeed the case for many rational functions phi.

  • 18. Krishnapur, Manjunath
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, Igor
    Nodal length fluctuations for arithmetic random waves2013In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 177, no 2, p. 699-737Article in journal (Refereed)
    Abstract [en]

    Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

  • 19.
    Kurlberg, Par
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lagarias, Jeffrey C.
    Pomerance, Carl
    On Sets of Integers Which Are Both Sum-Free and Product-Free2013In: Integers: Electronic Journal of Combinatorial Number Theory, ISSN 1553-1732, E-ISSN 1553-1732Article in journal (Refereed)
    Abstract [en]

    We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.

  • 20. Kurlberg, Pär
    A local Riemann hypothesis, II2000In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 233, no 1, p. 21-37Article in journal (Refereed)
  • 21.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Bounds on exponential sums over small multiplicative subgroups2007In: Additive Combinatorics / [ed] Grantville, A; Nathanson, MB; Solymosi, J, 2007, Vol. 43, p. 55-68Conference paper (Refereed)
    Abstract [en]

    We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang [6].

  • 22. Kurlberg, Pär
    Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps2007In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 8, no 1, p. 75-89Article in journal (Refereed)
    Abstract [en]

    We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck's constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are <<(epsilon) N-epsilon for all epsilon > 0 in the case of N square free.

  • 23. Kurlberg, Pär
    On a character sum problem of Cohn2002In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 92, no 1, p. 174-181Article in journal (Refereed)
    Abstract [en]

    Let f be a complex-valued function on a Finite field F such that f(0) = 0, f(1) = 1, and \f(x)\ = 1 for x not equal 0. H. Cohn asked if it follows that f is a nontrivial multiplicative character provided that Sigma(xis an element ofF) f(x) f(x+h) = -1 for h not equal 0. We prove that this is the case for finite Fields of prime cardinality under the assumption that the nonzero values of f are roots of unity.

  • 24. Kurlberg, Pär
    On the order of unimodular matrices modulo integers2003In: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 110, no 2, p. 141-151Article in journal (Refereed)
  • 25.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    POISSON SPACING STATISTICS FOR VALUE SETS OF POLYNOMIALS2009In: International Journal of Number Theory, ISSN 1793-0421, Vol. 5, no 3, p. 489-513Article in journal (Refereed)
    Abstract [en]

    If f is a non-constant polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes Poissonian as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity.

  • 26. Kurlberg, Pär
    The distribution of spacings between quadratic residues, II2000In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 120, p. 205-224Article in journal (Refereed)
    Abstract [en]

    We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

  • 27.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lagarias, J. C.
    Pomerance, C.
    Product-free sets with high density2012In: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 155, no 2, p. 163-173Article in journal (Refereed)
  • 28.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lagarias, Jeffrey C.
    Pomerance, Carl
    The Maximal Density of Product-Free Sets in Z/nZ2013In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 4, p. 827-845Article in journal (Refereed)
    Abstract [en]

    This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab=c (mod n), with a, b, c being in the set. In a previous paper, we constructed an infinite sequence of integers (n(i))(i >= 1) and product-free sets S-i in Z/n(i)Z such that the density vertical bar S-i vertical bar/n(i) -> 1 as i -> infinity, where vertical bar S-i vertical bar denotes the cardinality of S-i. Here, we obtain matching, up to constants, upper and lower bounds on the maximal attainable density as n -> infinity.

  • 29.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Luca, F.
    Shparlinski, I. E.
    On the fixed points of the map x→xx modulo a prime2015In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 22, no 1, p. 141-168Article in journal (Refereed)
    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.

  • 30.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pomerance, C
    On the periods of the linear congruential and power generators2005In: Acta Arithmetica, ISSN 0065-1036, E-ISSN 1730-6264, Vol. 119, no 2, p. 149-169Article in journal (Refereed)
  • 31.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pomerance, Carl
    On a problem of Arnold: The average multiplicative order of a given integer2013In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 7, no 4, p. 981-999Article in journal (Refereed)
    Abstract [en]

    For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l(g) (p) as p <= x ranges over primes.

  • 32.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rosenzweig, Lior
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Superscars for arithmetic toral point scatterers2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 1, p. 329-360Article in journal (Refereed)
    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.

  • 33.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rosenzweig, Lior
    Rudnick, Zeev
    Matrix elements for the quantum cat map: fluctuations in short windows2007In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 20, no 10, p. 2289-2304Article in journal (Refereed)
    Abstract [en]

    We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck's constant. We show that if the length of the window is smaller than the square root of Planck's constant, but larger than the separation between distinct eigenphases, then the variance of this sum is proportional to the length of the window, with a proportionality constant which coincides with the variance of the individual matrix elements corresponding to Hecke eigenfunctions.

  • 34. Kurlberg, Pär
    et al.
    Rudnick, Z.
    Hecke theory and equidistribution for the quantization of linear maps of the torus2000In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 103, no 1, p. 47-77Article in journal (Refereed)
  • 35. Kurlberg, Pär
    et al.
    Rudnick, Z.
    On quantum ergodicity for linear maps of the torus2001In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 222, no 1, p. 201-227Article in journal (Refereed)
    Abstract [en]

    We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (cat maps). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence., all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group. there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than rootN.

  • 36.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rudnick, Z.
    On the distribution of matrix elements for the quantum cat map2005In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 161, no 1, p. 489-507Article in journal (Refereed)
    Abstract [en]

    For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. In this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map. We present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture.

  • 37. Kurlberg, Pär
    et al.
    Rudnick, Z.
    Value distribution for eigenfunctions of desymmetrized quantum maps2001In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 18, p. 985-1002Article in journal (Refereed)
  • 38.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rudnick, Zeev
    The fluctuations in the number of points on a hyperelliptic curve over a finite field2009In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 129, no 3, p. 580-587Article in journal (Refereed)
    Abstract [en]

    The number of points on a hyperelliptic curve over a field of q elements may be expressed as q + 1 + S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that S/root q is distributed as the trace of a random 2g x 2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values 1 with probabilities 1/2(1 + q(-1)) and the value 0 with probability 1/(q + 1). When both the genus and the finite field grow, we find that S/root q has a standard Gaussian distribution.

  • 39.
    Kurlberg, Pär
    et al.
    Raymond/Beverly Sackler Sch. of M., Tel Aviv University.
    Rudnick, Zeév
    The distribution of spacings between quadratic residues1999In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 100, no 2, p. 211-242Article in journal (Refereed)
  • 40.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ueberschaer, Henrik
    Quantum Ergodicity for Point Scatterers on Arithmetic Tori2014In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 24, no 5, p. 1565-1590Article in journal (Refereed)
    Abstract [en]

    We prove an analogue of Shnirelman, Zelditch and Colin de VerdiS- re's quantum ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.

  • 41.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ueberschär, H.
    Superscars in the Šeba billiard2017In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 10, p. 2947-2964Article in journal (Refereed)
    Abstract [en]

    We consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes: "old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along four wave vectors in momentum space-we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit [5]. This result contrasts with the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus [14]. Further, in the strong coupling limit we show that a weaker form of localization holds for an essentially full density subsequence of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof andWinn who proved the existence of localized semiclassical measures under a clustering condition on the spectrum of the Laplacian.

  • 42.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ueberschär, Henrik
    Superscars in the Seba billiard2017In: 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)
  • 43.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, I.
    On probability measures arising from lattice points on circles2016In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, p. 1-42Article in journal (Refereed)
    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.

  • 44.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, Igor
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gaussian Point Count Statistics for Families of Curves Over a Fixed Finite Field2011In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, no 10, p. 2217-2229Article in journal (Refereed)
    Abstract [en]

    We produce a collection of families of curves, whose point count statistics over becomes Gaussian for p fixed. In particular, the average number of points on curves in these families tends to infinity.

  • 45.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, Igor
    Non-universality of the Nazarov-Sodin constant2015In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 353, no 2, p. 101-104Article in journal (Refereed)
    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.

  • 46.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wigman, Igor
    King's College London.
    Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 330, p. 516-552Article in journal (Refereed)
    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.

1 - 46 of 46
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