On numbers $n$ dividing the $n$th term of a linear recurrence
Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.
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arXiv subjects
Igor Shparlinski
Publications and source records attributed to Igor Shparlinski.
On Small Solutions to Quadratic Congruences
We estimate the deviation of the number of solutions of the congruence $$ m^2-n^2 \equiv c \pmod q, \qquad 1 \le m \le M, \ 1\le n \le N, $$ from its expected value on average over $c=1, ..., q$. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function $αk^2$, $k=1,2,...$ with a real $α$.
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Some Additive Combinatorics Problems in Matrix Rings
We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.
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On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators
In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degreegrowth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.
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On curves over finite fields with Jacobians of small exponent
We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1. We also show when g=1 or g=2 that our bounds are best possible.
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On Some Weighted Average Values of L-functions
Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $ε> 0$, and $q^ε \le N \le q^{1/2 -ε}$, $$ \sum_{χ\ne χ_0} |\sum_{n=1}^N χ(n)|^2 |L(1, χ)|^2 = (1 + o(1)) α_q q N $$ where the sum is take over all nonprincipal characters $χ$ modulo $q$, $L(s, χ)$ is the $L$-functions $L(1, χ)$ corresponding to $χ$ and $α_q = q^{o(1)}$ is some explicit function of $q$. Here we show that the same formula holds in the range $q^ε \le N \le q^{1 -ε}$.
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On the Sum-Product Problem on Elliptic Curves
Let $\E$ be an ordinary elliptic curve over a finite field $\F_{q}$ of $q$ elements and $x(Q)$ denote the $x$-coordinate of a point $Q = (x(Q),y(Q))$ on $\E$. Given an $\F_q$-rational point $P$ of order $T$, we show that for any subsets $\cA, \cB$ of the unit group of the residue ring modulo $T$, at least one of the sets $$ \{x(aP) + x(bP) : a \in \cA, b \in \cB\} \quad\text{and}\quad \{x(abP) : a \in \cA, b \in \cB\} $$ is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures.
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Parameters of Integral Circulant Graphs and Periodic Quantum Dynamics
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a circulant graph, is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus on relevant properties of integral circulant graphs. Specifically, we bound the number of vertices of integral circulant graphs in terms of their degree, characterize bipartiteness and give exact bounds for their diameter. Additionally, we prove that circulant graphs with odd order do not allow perfect state transfer.
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On a Generalisation of a Lehmer Problem
We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds of sums of multiplicative characters instead of traditionally applied to this kind of problem Kloosterman sums, we improve their results in several directions.
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On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.
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Complexity of Inverting the Euler Function
We present an algorithm to invert the Euler function $ϕ(m)$. The algorithm, for a given $n \geq 1$, in polynomial time ``on average'', finds the set $Ψ(n)$ of all solutions $m$ to $ϕ(m) = n$. In fact, in the worst case, $Ψ(n)$ is exponentially large, and cannot be computed in polynomial time. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that there is a polynomial time reduction of the Partition Problem, an NP-complete problem, to the problem of deciding whether $ϕ(m) = n$ has a solution for a small set of integers n. This shows that the problem of deciding whether a given finite set of integers S contains a totient is NP-complete. A totient is an integer n that lies in the image of the phi function; that is, an integer n for which there exists an integer m solving phi(m) = n. Finally, we establish close links between of inverting the Euler function and the integer factorization problem.
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Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation
We consider the problem of recovering a hidden monic polynomial f(X) of degree d > 0 over the finite field F of p elements given a black box which, for any x in F, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d^2 p^{d + c}), for any c > 0, and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X + s (with unknown s); some are new even in this case.
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