Abstract
Given a nondecreasing sequence \(\Lambda =\{\lambda _n>0\}\) such that \(\displaystyle \lim _{n\rightarrow \infty } \lambda _n=\infty ,\) we consider the sequence \(\mathcal {N}_\Lambda :=\left\{ \lambda _ne^{i\theta _n},n\in \,\mathbb {N}\right\} \), where \(\theta _n\) are independent random variables uniformly distributed on \([0,2\pi ].\) We discuss the conditions on the sequence \(\Lambda \) under which \(\mathcal N_\Lambda \) is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence \(\Lambda \) with respect to the weight is found.
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Acknowledgements
I am sincerely grateful to Alexander Borichev, who suggested the question studied in this paper and on several occasions helped with the proofs. I would also like to express my gratitude to Evgeny Abakumov and Mikhail Sodin, who have read preliminary version of this work and made helpful comments. I’m also grateful to the referee, who carefully read the paper and made many remarks and suggestions, which helped to improve the exposition.
The main results of this work (Sections 2, 3, 4) were obtained while the author was supported by Russian Science Foundation grant No. 20-61-46016. On the final stage of preparation of this paper the author was supported by Israel Science Foundation Grant 1288/21.
Funding
The main results of this work (Sections 2, 3, 4) were obtained while the author was supported by Russian Science Foundation grant No. 20-61-46016. On the final stage of preparation of this paper the author was supported by Israel Science Foundation Grant 1288/21.
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Appendix
Appendix
Here we recall some notions and results which we have used in this note.
1.1 Bernstein-Hoeffding concentration inequality [14, Theorem 2.2.6]
Let \(X_{1},X_{2},\dots ,X_{N}\) be independent random variables such that \({\displaystyle a_{i}\le X_{i}\le b_{i}}\) for every i. Then, for any \(t>0\) we have
1.2 Khinchine-Kolmogorov theorem on convergence of random series [1, Theorem 22.6]
Let \((Y_k)_{k\in {\mathbb {N}}}\) be a sequence of independent real-valued random variables with zero expectation. If
then the series \(\displaystyle \sum Y_k\) converges a.s.
1.3 Kolmogorov maximal inequality [1, Theorem 22.4]
Let \(Y_1, Y_2,\ldots , Y_N\) be a sequence of independent real-valued random variables with zero expectation and finite variances. For \(\gamma >0\)
1.4 Weyl-type criterion of uniform distribution of random sequence [6, Theorem 2]
Let \(X_{1},X_{2},\ldots \) be a sequence of independent real-valued random variables with characteristic functions \(\phi _1,\phi _2,\ldots \). Then this sequence is uniformly distrubuted modulo 1 almost surely, that is
with probability one, if and only if for every \(k\in {\mathbb {N}}\)
1.5 Definition of a set of disks with zero linear density [11, Chapter II, §1]
A set D of disks \(D_j\) in the complex plane is said to have zero linear density, if
where \(\lambda _j\) is the radius and \(c_j\) is the center of \(D_j\).
1.6 Levin-Pfluger theorem I [11, Chapter II, §1, Theorem 1]
Let a discrete set \({\mathcal {N}}\subset {\mathbb {C}}\) have an angular density of index \( \rho (r)\), where \(\rho (r)\) is a proximate order with \(\rho =\lim _{r\rightarrow \infty }\rho (r)\notin {\mathbb {N}}\). Let \(\Delta \) be a nondecreasing function such that for all but a countable set of angles
Then for \(z\in {\mathbb {C}}\setminus E\), where E is a set of disks with zero linear density, the canonical product
satisfies the asymptotic relation
1.7 Levin-Pfluger theorem II [11, Chapter II, §1, Theorem 2]
Let a set \({\mathcal {N}}\subset {\mathbb {C}}\) have an angular density of index \(\mathcal \rho (r)\), where \(\rho (r)\) is a proximate order with \(\rho =\lim _{r\rightarrow \infty }\rho (r)\in {\mathbb {N}}\). Let \(\Delta \) be a nondecreasing function such that for all but a countable set of angles
Let the following limits exist and be finite
Then for \(z\in {\mathbb {C}}\setminus E\), where E is a set of disks with zero linear density, the entire function
satisfies the following asymptotic relation
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Kononova, A. Random zero sets for Fock type spaces. Anal.Math.Phys. 13, 9 (2023). https://doi.org/10.1007/s13324-022-00770-x
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DOI: https://doi.org/10.1007/s13324-022-00770-x