On the number of polynomials of small house

Suppose that ɛ is a positive number, and letd be a sufficiently large positive integer. We consider a set of monic polynomials of degreed with integer coefficients and roots lying in the disk of radiusd ^ɛ/d. We prove that the number of such polynomials is less than exp(d ^2/3+ɛ). Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 214–219, April–June, 1999.     

Integer parts of powers of Pisot and Salem numbers

Let a \alpha be a Salem number or a Pisot number. We prove that integer parts of its powers [aⁿ] [\alpha^n] are composite for infinitely many integers n.     

Certain Diophantine Properties of the Mahler Measure

It is proved that a polynomial in several Mahler measures with positive rational coefficients is equal to an integer if and only if all these Mahler measures are integers. An estimate for the distance between a metric Mahler measure and an integer is obtained. Finally, it is proved that the ratio of two distinct Mahler measures of algebraic units is irrational.     

On the number of reducible polynomials of bounded naive height

We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\) . The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\) , where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\) . For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\) . Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba.     

Polynomials with high multiplicity at unity and Tarry’s problem

The existence of a polynomial with integer coefficients of relatively small length and sufficiently large multiplicity at unity is established. The proof of the corresponding statement is based on an estimate of the number of solutions of Tarry’s system. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 810–815, June, 1999.     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

There are only finitely many distance-regular graphs of fixed valency greater than two

In this paper we prove the Bannai–Ito conjecture, namely that there are only finitely many distance-regular graphs of fixed valency greater than two.     

On <Emphasis Type="Italic">β</Emphasis>-expansions of unity for rational and transcendental numbers <Emphasis Type="Italic">β</Emphasis>

We investigate the sequence of integers x ₁, x ₂, x ₃, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = \(\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } \) for rational and transcendental numbers β > 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj) _j=1 ^∞ in terms of β.