On Crossings of Hyperplanes on then-Dimensional Torus Original Research Article

LetSbe a set of hyperplanes of then-dimensional Euclidean spaceRn. For each (r1,…,rn) set membership, variant Rn, let [r1,…,rn] = {(s1,…,sn) set membership, variant Rnmidsi − riis an integer for each 1 ≤ i ≤ n}. For eachH set membership, variant S, letimage. In this paper, we study the set intersection operatorH set membership, variant S imagewhenever eachH set membership, variant Ssatisfies a linear equation with rational coefficients of all nonconstant terms.