| 1. |
- Cammarota, Valentina, et al.
(author)
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Boundary Effect on the Nodal Length for Arithmetic Random Waves, and Spectral Semi-correlations
- 2020
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In: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 376:2, s. 1261-1310
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Journal article (peer-reviewed)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.
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| 2. |
- Huang, Hao, et al.
(author)
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On subsets of the hypercube with prescribed Hamming distances
- 2020
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In: Journal of combinatorial theory. Series A (Print). - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0097-3165 .- 1096-0899. ; 171
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Journal article (peer-reviewed)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.
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| 3. |
- Klurman, Oleksiy, et al.
(author)
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On the orbits of multiplicative pairs
- 2020
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In: Algebra & Number Theory. - : MATHEMATICAL SCIENCE PUBL. - 1937-0652 .- 1944-7833. ; 14:1, s. 155-189
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Journal article (peer-reviewed)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.
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| 4. |
- Kurlberg, Pär, et al.
(author)
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A note on multiplicative automatic sequences, II
- 2020
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In: Bulletin of the London Mathematical Society. - : London Mathematical Society. - 0024-6093 .- 1469-2120. ; 52:1, s. 185-188
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Journal article (peer-reviewed)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].
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