Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — arf < c, r(q — fir) < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals a and /3, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46"" = 1/2.6043 • • •.

Let Pd denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers x1 ≤ x2 ≤ . . . is called Pd-controlling if there exist y1, y2, . . . ∈ R such that for every polynomial p ∈ Pd there exists an index i with |p(xi)−yi| ≤ 1. We settle an problem of Makai and Pach (1983) by showing that x1 ≤ x2 ≤ . . . is Pd-controlling if and...

In a classical problem in scheduling, one has n unit size jobs with a precedence order and the goal is to find a schedule of those jobs on m identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is NP-hard form = 3. We prove that for any fixed ε and m, a Sherali-Adams / Lasserre lift of the time...

This paper proves a tradeoff between the time it takes to search for elements in an implicit dictionary and the time it takes to update the value of elements in specified locations of the dictionary. It essentially shows that if the update time is constant, then the search time is $\Omega(n’)$ for some constant $\varepsilon >0$.

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...

Hildebrand proved that the smooth approximation for number $\varPsi (x,y)$ of $y$-friable integers not exceeding $x$ holds $y \gt (\log x)^{2+\varepsilon }$ under Riemann hypothesis and he conjectured it fails when $y\leqslant x)^{2