Difference sets and inverting the difference operator
For \(A\subseteq N_0\) let \(D(A)\) be the set of differences of elements of \(A\) in \(N_0\). Problem: study equations \(D^k(X)=B\). There is a solution if \(0\in B\) and for each \(n\) there exists \(x\geq n\) such that each of the \(2^{k-1}\) intervals \([x-n, x+n]\), \([2x-n, 2x+n],\dots,[2^{k-1}x-n, 2^{k-1}x+n]\) is contained in \(B\). Such \(B\) are ``many'' (in various senses, e.g., for each \(k\) the family of \(B\)'s is of measure 1), but very ``big''. Call \(A\) a \(B_h\) set if distinct multisets of \(h\) elements of \(A\) always have distinct sums. Now the equation \(D^k(X)=B\) has only a solution (obviously modulo translations) if a solution is a \(B_h\) set with \(h=2^{2k-1}+2^{k-1}\). Cases in which \(D^k(X)=B\) has exactly \(2^t\) solutions (for suitable \(t\)) are given. The paper is complex, uses ideas borrowed from linear algebra and hypergraph theory, and ends with interesting open problems.
scientific article; zbMATH DE number 878264