A Simplified Proof of the Erdos-Fuchs Theorem

Abstract

We reprove the theorem of Erdos and Fuchs in additive number theory. Whereas their solution rested on some special results in the L2 theory of Fourier series, ours avoids these. We present a variant of the proof of the very pretty theorem of Erdos and Fuchs [1]. Our proof is technically a bit simpler than theirs but, of more importance, it has the aesthetic advantage of sticking closer to the spirit of generating functions. THEOREM (ERD6s-FucHs). Let A be a set of nonnegative integers and denote by r(n) the number of solutions to n = a + a', a, a' E A. If for some C > 0, n -0(r(k) -C) = O(na), then a PROOF. If we write An = X70O(r(k)-C) then we have (Za)2= + (1 z)2AnZn, An = O(n ), (1) Here, as later, we abbreviate our summation notation. It is to be generally understood that a ranges over the set A, that n ranges over the nonnegative integers, and, when we use the letter b, it will range over the nonnegative integers below N. So let us multiply (1) by (1 + z + z2 + + zN-1)2, N > 1, and obtain thereby ( Z a ) C( zb)2ZN b EA (2) which in turn gives the inequality IEZ aE bI2 2 rn X r2b. (4) Received by the editors January 27, 1978. AMS (MOS) subject classifications (1970). Primary IOA45, IOL05.