Lagrange ' S Theorem with N " 3 Squares

For every N > I we construct a set A of squares such that JA < (4/log 2)N 1 / 3 log N and every nonnegative integer n < N is a sum of four squares belonging to A . Let A be an increasing sequence of nonnegative integers and let A(x) denote the number of elements of A not exceeding x. If every nonnegative integer up to x is a sum of four elements of A, then A(x)4 > x and so A(x) > x 1/4. In 1770, Lagrange proved that every integer is a sum of four squares . If A is a subsequence of the squares such that every nonnegative integer is a sum of four squares belonging to A, then we say that Lagrange's theorem holds for A . Since there are 1 + [x 12] nonnegative squares not exceeding x, it is natural to look for subsequences A of the squares such that Lagrange's theorem holds for A and A is "thin" in the sense that A(x) < cx" for some a < 1/2 . Hartter and Zöllner [2] proved that there exist infinite sets S of density zero such that Lagrange's theorem holds for A = {n2In 1 S) . It is still true in this case that A(x) x1/2. Using probabilistic methods, Erdös and Nathanson [1] proved that, for every e > 0, Lagrange's theorem holds for a sequence A of squares satisfying A(x) < cx (3/8)+e In this paper we study a finite version of Lagrange's theorem . For every N > 1, we construct a set A of squares such that JA < (4/log 2)N 1 / 3 log N and every n < N is the sum of four squares belonging to A . This improves the result of Erdös and Nathanson in the case of finite intervals of integers . We conjecture that for every e > 0 and N > N(e) there exists a set A of squares such that JA < N (1 /4)+e and every n < N is the sum of four squares in A . Let JA denote the cardinality of the finite set A and let [x] denote the greatest integer not exceeding x . LEMMA. Let a > 1 . Let n > a2 and n 5~ 0 (mod 4) . Then either n a2 or n (a 1)2 is a sum of three squares . PROOF . If the positive integer m is not a sum of three squares, then m is of the form m = 4s(8t + 7). If s = 0, then m 3 (mod 4) . If s > 1, then m 0 (mod 4) . Received by the editors May 11, 1979 and, in revised form, September 21, 1979 . 1980 Mathematics Subject Classification . Primary 1OJ05; Secondary 1OL05, IOL02, 1OL1O .