Generalized and restricted multiplication tables of integers

In 1955 Erd??s posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a???N and b???N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erd??s's question in two different directions. The first one concerns the k-dimensional version of the multiplication table problem: for a fixed integer k???3 and a large parameter N, we establish the order of magnitude of the number of distinct products n_1...n_k with n_i???N for all 1???i???k. The second question we shall discuss is the restricted multiplication table problem. More precisely, for a set of integers B we seek estimates on the number of distinct products ab in B with a???N and b???N. This problem is intimately connected with the available information on the distribution of B in arithmetic progressions. We focus on the special and important case when B={p+s:p prime} for some fixed non-zero integer s. Ford established upper bounds of the expected order of magnitude for |{p+s=ab:a???N,b???N}|. We prove the corresponding lower bounds thus determining the size of the quantity in question up to multiplicative constants