Lonely Points in Simplices.

Given a lattice L ⊆ Z m and a subset A ⊆ R m , we say that a point in A is lonely if it is not equivalent modulo L to another point of  A . We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.