arXiv subjects
Publications and source records attributed to Alex V. Kontorovich.
Selberg identified the "parity" barrior, that sieves alone cannot distinguish between integers having an even or odd number of factors. We give here a short and self-contained demonstration of parity breaking using bilinear forms, modeled on the Twin Primes Conjecture.
This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem. The stochastic models are rigorously analyzable, and yield heuristic predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.
We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.
The (3x+1)-Map, T, acts on the set, Pi, of positive integers not divisible by 2 or 3. It is defined by T(x) = (3x+1)/2^k, where k is the largest integer for which T(x) is an integer. The (3x+1)-Conjecture asks if for every x in Pi there exists an integer, n, such that T^n (x) = 1. The Statistical (3x+1)-Conjecture asks the same question, except for a subset of Pi of density 1. The Structure Theorem proven in \cite{sinai} shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x+1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x+1)-Map, and expand on the consequences derived in \cite{sinai}. The generalizations we consider are determined by positive coprime integers, d and g, with g > d >= 2, and a periodic function, h(x). The map T is defined by the formula T(x) = (gx+h(gx))/d^k, where k is again the largest integer for which T(x) is an integer. We prove an analogous Structure Theorem for (d,g,h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift log(g) - d/(d-1)log(d) and positive diffusion constant. This shows that it is natural to expect that typical trajectores return to the origin if log(g) - d/(d-1) log(d) <0 and escape to infinity otherwise.
We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford.
For every positive integer $n$, the quantum integer $[n]_q$ is the polynomial $[n]_q = 1 + q + q^2 + ... + q^{n-1}.$ A quadratic addition rule for quantum integers consists of sequences of polynomials $\mathcal{R}' = \{r'_n(q)\}_{n=1}^{\infty}$, $\mathcal{S}' = \{s'_n(q)\}_{n=1}^{\infty}$, and $\mathcal{T}' = \{t'_{m,n}(q)\}_{m,n=1}^{\infty}$ such that $[m+n]_q = r'_n(q)[m]_q + s'_m(q)[n]_q + t'_{m,n}(q)[m]_q[n]_q$ for all $m$ and $n.$ This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials \polf that satisfy the associated functional equation $f_{m+n}(q)= r'_n(q)f_m(q) + s'_m(q)f_n(q) + t'_{m,n}f_m(q)f_n(q).$