This talk surveys number theory that appeared in my research on analysis. We shall first introduce the Erdos distance problem in which the integer lattice serves as the sharp example. Then we will discuss its continuous version, the Falconer distance problem, in which the sharp example is given by Jarnik's construction based on diophantine approximation. In the second half of this talk we shall introduce tiling and multiple tiling by translations in the Euclidean space, and explain how counting solutions of linear equations in the multiplicative group influences our result on the multiplicity of tiling.