| 1. |
- Cammarota, Valentina, et al.
(författare)
-
Boundary Effect on the Nodal Length for Arithmetic Random Waves, and Spectral Semi-correlations
- 2020
-
Ingår i: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 376:2, s. 1261-1310
-
Tidskriftsartikel (refereegranskat)abstract
- We test M. Berry’s ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions (“boundary-adapted arithmetic random waves”). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac–Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of “bad” subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the “spectral semi-correlations”, a concept joining the likes of “spectral correlations” and “spectral quasi-correlations” in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.
|
|
| 2. |
- Huang, Hao, et al.
(författare)
-
On subsets of the hypercube with prescribed Hamming distances
- 2020
-
Ingår i: Journal of combinatorial theory. Series A (Print). - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0097-3165 .- 1096-0899. ; 171
-
Tidskriftsartikel (refereegranskat)abstract
- A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in {0, 1}(n) with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetkovid bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers of width much smaller than n. We also improve on a theorem of Alon about subsets of F-p(n) whose difference set does not intersect {0, 1}(n) nontrivially.
|
|
| 3. |
- Klurman, Oleksiy, et al.
(författare)
-
A note on multiplicative automatic sequences
- 2019
-
Ingår i: Comptes rendus. Mathematique. - : ELSEVIER FRANCE-EDITIONS SCIENTIFIQUES MEDICALES ELSEVIER. - 1631-073X .- 1778-3569. ; 357:10, s. 752-755
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 4. |
- Klurman, Oleksiy, et al.
(författare)
-
On the orbits of multiplicative pairs
- 2020
-
Ingår i: Algebra & Number Theory. - : MATHEMATICAL SCIENCE PUBL. - 1937-0652 .- 1944-7833. ; 14:1, s. 155-189
-
Tidskriftsartikel (refereegranskat)abstract
- We characterize all pairs of completely multiplicative functions fg : N -> T, where T denotes the unit circle, such that <({f(n), g(n+1))}(n >= 1))over bar> not equal T x T. In so doing, we settle an old conjecture of Zoltan Daroczy and Imre Katai.
|
|
| 5. |
- Klurman, Oleksiy, et al.
(författare)
-
Rigidity theorems for multiplicative functions
- 2018
-
Ingår i: Mathematische Annalen. - : Springer New York LLC. - 0025-5831 .- 1432-1807. ; 372:1-2, s. 651-697
-
Tidskriftsartikel (refereegranskat)abstract
- We establish several results concerning the expected general phenomenon that, given a multiplicative function f: N→ C, the values of f(n) and f(n+ a) are “generally” independent unless f is of a “special” form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function f: N→ T we have lim infn→∞|f(n+1)-f(n)|=0.Second, we settle an old conjecture due to Chudakov (On the generalized characters. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, p. 487. Gauthier-Villars, Paris) that states that any completely multiplicative function f: N→ C that: (a) takes only finitely many values, (b) vanishes at only finitely many primes, and (c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erdős Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character χ mod q then f(n) = χ′(n) nit for all n, where χ′ is a Dirichlet character modulo q and t∈ R. This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for pretentious multiplicative functions developed earlier by the first author and Szemeredi’s theorem for long arithmetic progressions.
|
|
| 6. |
- Klurman, Oleksiy, et al.
(författare)
-
V. Markov's problem for k-absolutely monotone polynomials and applications
- 2019
-
Ingår i: Jaen Journal on Approximation. - : Universidad de Jaen. - 1889-3066 .- 1989-7251. ; 11:1-2, s. 139-149
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the classical problem of maximizing the value of the derivative of a polynomial at a given point x0 j [-1,1]. The corresponding extremal problem for general polynomials in the uniform norm was solved by V. Markov. In this paper, we consider the analog of this problem for k-bsolutely monotone polynomials. As a consequence, we solve the analog of V. Markov' problem, find the exact constant in Bernstein' inequality and give a new proof of A. Markov' inequality for monotone polynomials.
|
|
| 7. |
- Kurlberg, Pär, et al.
(författare)
-
A note on multiplicative automatic sequences, II
- 2020
-
Ingår i: Bulletin of the London Mathematical Society. - : London Mathematical Society. - 0024-6093 .- 1469-2120. ; 52:1, s. 185-188
-
Tidskriftsartikel (refereegranskat)abstract
- We prove that any q'>?q‐automatic multiplicative function?:ℕ→ℂ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of Bell, Bruin and Coons [Trans. Amer. Math. Soc. 364 (2012) 933–959].
|
|
