Following a statement of the well-known Erdos-Turán conjecture,
Erdos mentioned the following even stronger conjecture: if the
n-th term
an of a sequence
A of positive integers is bounded by
![$\alpha n^2$](img2.gif)
,
for some positive real constant
![$\alpha$](img3.gif)
,
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
an
differs from
![$\alpha n^2$](img2.gif)
(or from a quadratic polynomial with rational
coefficients
q(
n)) by at most
![$o (\sqrt {\log n})$](img4.gif)
,
then the number of
representations function is indeed unbounded.