Ruzsa-type inequalities for additive energy - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:17:43Z http://mathoverflow.net/feeds/question/93545 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93545/ruzsa-type-inequalities-for-additive-energy Ruzsa-type inequalities for additive energy Cosmin Pohoata 2012-04-09T05:38:32Z 2012-04-09T11:22:07Z <p>As the subject says, I have some questions about some Ruzsa-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.</p> <p>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 $F(A,B)= \frac{|A|^{2}|B|^{2}}{E(A,B)}$.</p> <p>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?</p> <p>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 $c$.</p> <p>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 $F(k \cdot A, l \cdot A) \leq K^{c} N$.</p> http://mathoverflow.net/questions/93545/ruzsa-type-inequalities-for-additive-energy/93555#93555 Answer by Boris Bukh for Ruzsa-type inequalities for additive energy Boris Bukh 2012-04-09T09:17:03Z 2012-04-09T09:27:51Z <p>1) No. Let $A=B=C$ be a set that is a union of interval <code>$\{1,\dots,N\}$</code> and $N$ random elements from <code>$\{1,\dotsc,N^2\}$</code>. Then $F(A,A)=KN$ for some constant $K$. On the other hand, $A+A$ is basically the interval <code>$\{1,\dotsc,N^2\}$</code>. So, $F(A,A+A)\approx N^2$. </p> <p>2) Example above show that it fails for $k=1$ and $l=2$ (from the context I assume that $k\cdot A$ means $k$-fold sumset $A+\dotsb+A$, not the $k$-dilate <code>$\{ka:a\in A\}$</code>. If dilates were meant, then the assertion is true, and one can deduce this from the Balog–Szemerédi–Gowers theorem).</p> <p>I want to add that there do exist Ruzsa-type inequalities for additive energy. Instead of sumsets, they involve a suitable extension of additive energy to more than two sets. See <a href="http://people.cs.uchicago.edu/~razborov/files/free_group.pdf" rel="nofollow">Razborov's paper on the product theorem in the free group</a> (in section 6), for example.</p>