On the distribution of multiples of real numbers

We introduce new series (of the variable ??) that enable to measure the irregularity of distribution of the sequence of fractional parts {n??}. A detailed analysis of the convergence and divergence of these series is done, depending mainly on the convergents of ??. As a by product, we obtain new Fourier series of square integrable functions that converge almost everywhere but at no rational number.     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

On digit sums of multiples of an integer

Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.     

On weighted densities

The continuity of densities given by the weight functions n ^α , α ∈ [−1, ∞[, with respect to the parameter α is investigated. This work is supported by MIUR Italy, Program Barrande n. 2003-009-2, MSM6198898701 and GA ČR no. 201/04/0381.     

On existence of common multiples of monic operator polynomials

A sufficient condition (in terms of spectral behaviour) for existence of common multiples of monic operator polynomials is given.     

On the spiral-likeness of rational functions

The object of this paper is to extend some results concerning the univalence, starlikeness, and convexity of rational functions recently obtained by Reade, Silverman, and Todorov. The domain of variability of log{f(z)/z} for a fixedz and for such functionsf ranging over the class of λ-spirallike functions of order α are also determined.     

On four-colourings of the rational four-space

Summary LetQ ⁴ denote the graph, obtained from the rational points of the 4-space, by connecting two points iff their Euclidean distance is one. It has been known that its chromatic number is 4. We settle a problem of P. Johnson, showing that in every four-colouring of this graph, every colour class is every-where dense.     

On calculation of generalized densities

In the paper continuous variants of densities of sets of positive integers are introduced, some of their properties are studied and formula for their calculation is proved. Supported by the grant GA CR no. 201/04/0381.     

On the pointwise densities of the Cantor measure

Let K(a) be the symmetrical Cantor set generated by ?₀(x)=ax and ?₁(x)=ax+(1−a), where 0<a<1/2. Let s be the Hausdorff dimension of K(a) and μ the Cantor measure. In this paper, under the hypothesis that a is slightly greater than 1/3, we obtain the explicit formulas of the upper and lower s-densities Θ?s(μ,x), for every point x∈K(a). Moreover, we describe the range of a key quantity τ(x) in these formulas.     

On the discrepancy of the sequence formed by the multiples of an irrational number

Let θ be an irrational number, and consider sequences of the form ω_θ = 〈kθ〉_k≥0 of points in the circle R/Z. By employing symmetry, we can show that the discrepancy D_N(ω_θ) of the finite sequence 〈kθ〉_0≤k<N is determined by its behavior on the N arcs whose endpoints are iθ and (N ? 1 ? i)θ for 0 ≤ i < N. We then use continued fraction methods to analyze its behavior on these arcs. The resulting expression for D_N(ω_θ has several consequences. First, we show that the discrepancies D_N(ω_σ) and D_N(ω_τ) are closely related if σ and τ are equivalent irrationals; in particular, we prove the equality $\text{lim sup}{}_{\mathrm{N}}\text{(}\text{ND}{}_{\mathrm{N}}\text{(ω}{}_{\mathrm{\sigma }}\text{)}\text{log}\text{N}\text{) =}\text{lim sup}{}_{\mathrm{N}}\text{(}\text{ND}{}_{\mathrm{N}}\text{(ω}{}_{\mathrm{\tau }}\text{)}\text{log}\text{N}\text{)}$. Finally, we compute a tight asymptotic bound on D_N(ω_θ) when θ has the special form $\text{θ =}\text{(√m}{}^2\text{+ 4 ? m)}\text{2}$ for some positive integer m by showing that

$\text{lim}{}_{\mathrm{N}}\text{sup}\text{ND}{}_{\mathrm{n}}\text{(ωθ)}\text{log N}\text{=}\text{{}\text{m}\text{4 log(}\text{1}\text{θ}\text{)}\text{if m is even,}\text{(}\text{m}{}^2\text{+3}\text{m}{}^2\text{+4}\text{)}\text{n}\text{4 log (}\text{1}\text{θ}\text{)}\text{if m is odd.}$