Abstract: We present efficient algorithms for the on-line $q$-adic covering of the unit interval by sequences of segments. The basic method guarantees covering provided the total length of segments is at least $1+ 2\cdot {1\over q} - {1\over q^3}$. Other algorithms improve this estimate for $q\geq 6$. The unit $d$-dimensional cube can be on-line covered by an arbitrary sequence of cubes whose total volume is at least $2^d+{5\over 3}+{5\over 3}\cdot 2^{-d}$.
Classification (MSC2000): 52C17
Full text of the article: