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4 fix spelling of Imre Ruzsa

# Rusza-typeRuzsa-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 $F(k \cdot A, l \cdot A) \leq K^{c} N$.

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$. 2 deleted 1 characters in body As the subject says, I have some questions about some Rusza-type inequalities for additive-energy. It would be wonderful if anyone could shed some light. First, the definition: let$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\$.

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