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Representation of Integers by Near Quadratic
Sequences Labib Haddad
120 rue de Charonne
75011 Paris
France
mailto:labib.haddad@wanadoo.frlabib.haddad@wanadoo.fr

Charles Helou
Department of Mathematics
Pennsylvania State University
25 Yearsley Mill Road
Media, PA 19063
USA
mailto:cxh22@psu.educxh22@psu.edu

in 0=+ =0 0=- =0 0=$\cdots$ =

Abstract:

Following a statement of the well-known Erdos-Turán conjecture, Erdos mentioned the following even stronger conjecture: if the n-th term a_n of a sequence A of positive integers is bounded by $\alpha n^2$, for some positive real constant $\alpha$, then the number of representations of n as a sum of two terms from A is an unbounded function of n. Here we show that if a_n differs from $\alpha n^2$ (or from a quadratic polynomial with rational coefficients q(n)) by at most $o (\sqrt {\log n})$, then the number of representations function is indeed unbounded.