show/hide this revision's text 4 fix spelling of Imre Ruzsa

Rusza-type Ruzsa-type inequalities for additive energy

As the subject says, I have some questions about some Rusza-type Ruzsa-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.

First, the definition: let $E(A,B)$ be the number of all quadruples $(a,a',b,b') \in A \times A \times B \times B$ such that $a+b = a' + b'$ denote the additive energy of the sets $A$, $B$, and let $F(A,B)= \frac{|A|^{2}|B|^{2}}{E(A,B)}$.

Now let $A, B, C \subset G$ be subsets of an additive group $G$ so that $|A|=|B|=|C|=N$. Do the following inequalities hold?

1) If $F(A,B), F(A,C) \leq K\cdot N$, then $\min{(F(A, B+C), F(B, C+A), F(C, A+B))} \leq K^{c} N$, for some constant $c$.

2) If $F(A,A) \leq K\cdot N$, then for all non-negative integers $k,l$ there's a constant $c$ depending on them such that $F(k \cdot A, l \cdot A) \leq K^{c} N$.
show/hide this revision's text 3 added 53 characters in body; deleted 3 characters in body

As the subject says, I have some questions about some Rusza-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.

First, the definition: let $E(A,B)$ be the number of all quadruples $(a,a',b,b') \in A \times A \times B \times B$ such that $a+b = a' + b'$ denote the additive energy of the sets $A$, $B$.B$, and let $F(A,B)= \frac{|A|^{2}|B|^{2}}{E(A,B)}$.

Now let $A, B, C \subset G$ be subsets of an additive group $G$ so that $|A|=|B|=|C|=N$. Do the following inequalities hold?

1) If $E(A,B), E(A,CF(A,B), F(A,C) \leq K\cdot N$, then $\min{(E(A, \min{(F(A, B+C), E(BF(B, C+A), E(CF(C, A+B))} \leq K^{c} N$, for some constant $c$.

2) If $E(A,A) F(A,A) \leq K\cdot N$, then for all non-negative integers $k,l$ there's a constant $c$ depending on them such that $E(k F(k \cdot A, l \cdot A) \leq K^{c} N$.
show/hide this revision's text 2 deleted 1 characters in body

As the subject says, I have some questions about some Rusza-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.

First, the definition: let $E(A,B)$ be the number of all quadruples $(a,a',b,b') \in A \times A \times B \times B$ such that $a+b = a' + b'$ denote the additive energy of the sets $A$, $B$.

Now let $A, B, C \subset G$ be subsets of an additive group $G$ so that $|A|=|B|=|C|=N$. Do the following inequalities hold?

1) If $E(A,B), E(A,C) \leq K\cdot N$, then $\min{(E(A, B+C), E(B, C+A), E(C, A+B))} \leq K^{c} N$, for some constant $c$.

2) If $E(A,A) \leq K\cdot N$, then for all non-negative integers $k,l$ there's a constant $c$ depending on them such that $E(k \cdot A- , l \cdot A) \leq K^{c} N$.
show/hide this revision's text 1