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J. Bourgain

Publications and source records attributed to J. Bourgain.

6 recordsLinked to original sources
arXiv2013

Kloosterman sums in residue rings

In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$ I^{-1}=\{x^{-1}:\quad x\in I\}, $$ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun-Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general modulus.

math.NT↗
arXiv2013

Quantum recurrence of a subspace and operator-valued Schur functions

A notion of monitored recurrence for discrete-time quantum processes was recently introduced in [Commun. Math. Phys., DOI 10.1007/s00220-012-1645-2] (see also arXiv:1202.3903) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in the above mentioned paper. The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time. This notion of subspace recurrence also links the concept of expected return time to an Aharonov-Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counterintuitive behavior of quantum systems. All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.

quant-ph↗
arXiv2012

Sumsets of reciprocals in prime fields and multilinear Kloosterman sums

We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.

math.NT↗
arXiv2011

On representation of integers by binary quadratic forms

Given a negative $D>-(\log X)^{\log 2-δ}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant $D$. We also give an analogous upper bound for square free integers of the form $q+a<X$ where $q$ is prime and $a\in\mathbb Z$ is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers $q+a<X$ represented by a binary quadratic form of discriminant $D$, where $D$ is allowed to grow with $X$ as above. An immediate consequence of this, coming from recent work of the authors in [BF], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.

math.NT↗