Let $A$ be an infinite set of positive integers, and denote by $r(n)$ the number of solutions to the equation $a+a'=n$, with $a,\, a' \in A$.

It is not very difficult to show that if $r(n) > 0$ for $n$ large enough, then $r(n)$ cannot be a constant for $n$ large enough.

A refinement of this fact is given by Erdös-Fuchs Theorem (P.Erdös and W.H.J. Fuchs, On a problem of additive number theory, J. London Math. Soc 31 (1956), 67-73): let $C$ be a constant. Then the following situation

$$\sum_{m\leq n}( r(m)- C)= o\left(n^{1/4}\log(n)^{-1/2}\right)$$

cannot hold (see also the paper of I. Ruzsa for a construction with $O\left(n^{1/4}\log(n)\right)$)

Now, consider other kind of representation functions. Again, for an infinite sequence $A$, write $r_{2,3}(n)$ the number of solutions to the equation $2a+3a'=n$, with $a,\,a'\in A$.

Question: Does $r_{2,3}(n)$ satisfy an Erdös-Fuchs type result?

This question was initially formulated by A. Sárközy and V. Sós in Problem 7.2, Section 7 of (A. Sarkozy, V. Sós , `On additive representation functions',The mathematics of Paul Erdös I, Algorithms Combin. 13, pp 129-150), and I do not know if there has been advances in order to answer it.


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