Smallest number of sensors for k-Covering

This paper presents some theoretical results on the smaller number N_k(a, b) of sensors to achieve k coverage for the rectangular area [0, a] × [0, b]. The first properties show the numbers N_k(a, b) are sub-additive and increasing on each variable. Based on these results, some lower and upper bounds for N_k(a, b) are introduced. The main result of the article proves that the minimal density of sensors to achieve k-coverage is λ(k) ≤ k/2 improving a previous result of Ammari and Das [2]. Finally, the numbers N₁(a, b) are tabled for some small values of a, b.