Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as the `Arithmetic Kakeya Conjecture'. We consider a higher dimensional variant of this problem and prove an upper bound using a certain iterative argument. The main new ingredient in our proof is a general way to strengthen the sum-difference inequalities of Katz and Tao which might be of independent interest. As a corollary, we obtain a new lower bound for the Minkowski dimension of $(n, d)$-Besicovitch sets.