Splitter Sets and $k$ -Radius Sequences

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict-avoiding codes. The study of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-radius sequences was motivated by some problems occurring in large data transfer. It is observed that the existence of splitter sets yields <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-radius sequences of short length. In this paper, we obtain several new results contributing to splitter sets and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-radius sequences. We give some new constructions of perfect splitter sets, as well as some nonexistence results on them. As a byproduct, we obtain some new results on optimal conflict-avoiding codes. Furthermore, we provide several explicit constructions of short <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-radius sequences for certain values of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, by establishing the existence of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-additive sequences. In particular, we show that for any fixed <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, there exist infinitely many values of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$f_{k}(n) =\frac {n^{2}}{2k}+O(n)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$f_{k} (n)$ </tex-math></inline-formula> denotes the shortest length of an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-radius sequence. This result partially affirms a conjecture posed by Bondy, Lonc, and Rzążewski.