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7 corrected the name of Csaba Sándor

The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.

Let $B \subseteq {\mathbb N}$. If, for any natural number $k$, we denote by $r_B(k)$ the number of pairs $(i,j)$ in $B\times B$ such that $i+j=k$.

We call $B$ an additive basis of order two if $r_B$ is never $0$.

Erdos-Turan Conjecture for order two bases: If $B$ is an additive basis of order 2, then $r_B$ is unbounded.

Are there any serious strategies for attacking this conjecture? If so, what are they?

Application of Szemeredi's theorem quickly handles sets $B$ of positive upper density. The interesting case is the zero upper density case.

The most recent thing I've seen on this is the paper

Sandor, Szaba Csaba A note on a conjecture of Erdos-Turan, INTEGERS: Electronic Journal of Combinatorial Number Theory 8 no. 1 (2008).

(This question may be better for mathstackexchange, but I'm curious if there are any developed lines of attack for research questions like this. Harebrained subquestion: Can Green-Tao type techniques be leveraged for this problem?)

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The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.

Let $B \subseteq {\mathbb N}$. If, for any natural number $k$, we denote by $r_B(k)$ the number of pairs $(i,j)$ in $B\times B$ such that $i+j=k$.

We call $B$ an additive basis of order two if $r_B$ is never $0$.

Erdos-Turan Conjecture for order two bases: If $B$ is an additive basis of order 2, then $r_B$ is unbounded.

Are there any serious strategies for attacking this conjecture? If so, what are they?

Application of Szemeridi's Szemeredi's theorem quickly handles sets $B$ of positive upper density. The interesting case is the zero upper density case.

The most recent thing I've seen on this is the paper

Sandor, Szaba A note on a conjecture of Erdos-Turan, INTEGERS: Electronic Journal of Combinatorial Number Theory 8 no. 1 (2008).

(This question may be better for mathstackexchange, but I'm curious if there are any developed lines of attack for research questions like this. Harebrained subquestion: Can Green-Tao type techniques be leveraged for this problem?)

5 edited body

The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.

Let $B \subseteq {\mathbb N}$. If, for any natural number $k$, we denote by $r_B(k)$ the number of pairs $(i,j)$ in $B\times B$ such that $i+j=k$.

We call $B$ an additive basis of order two if $r_B$ is never $0$.

Erdos-Turan Conjecture for order two bases: If $B$ is an additive basis of order 2, then $r_B$ is unbounded.

Are there any serious strategies for attacking this conjecture? If so, what are they?

Application of Szemeridi's theorem quickly handles sets $B$ of positive upper density. The interesting case is the zero upper density case.

The most recent thing I've seen on this is the paper

Sandor, Szaba A note on a conjecture of Erdos-Turan, INTEGERS: Electronic Journal of Combinatorial Number Theory 8 no. 1 (2008).

(This question may be better for mathstackexchange, but I'm curious if there are any developed lines of attack for research questions like this. Hairbrained Harebrained subquestion: Can Green-Tao type techniques be leveraged for this problem?)

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