Sum-product phenomenon

The sum-product phenomenon says that a finite subset of a field cannot usually have both a small sumset and a small product set unless it has additional subfield-like structure.

Write $D(t)=|A+tA|/|A|$ and suppose some nonzero $b$ has $D(b)\le K$. The mixed-set Plünnecke-Ruzsa inequality gives $|3A-2A|\le K^5|A|$ and $|A+A|\le K^2|A|$. The Ruzsa covering lemma covers $tA$ by at most $D(t)$ translates of $A-A$. Covering two dilates proves the addition bound. First covering $sA$, then both copies of $rA$ in $r(A-A)$, proves the multiplication bound. The Ruzsa triangle inequality also gives $D(-r)\le K^{2}D(r)$. These bounds transfer controlled additive expansion through a bounded-complexity expression in scalars.

For positive $\alpha ,\beta$, suppose $p^{\alpha }\le |A|\le p^{1-\alpha }$ and $|B|\ge p^{\beta }$. Repeatedly replace a scalar set $S$ by $3SS-3SS$. Quadratic growth of a sixfold product difference set reaches size at least $p/2$ in a number of steps depending only on $\beta$. The closure bounds for small skew-sumset scalars transfer a putative bound $|A+bA|\le K|A|$ to all these scalars with exponent depending only on the number of steps. Averaging collisions over the resulting large scalar set then forces $K^L\ge \frac{1}{4}\min (|A|,p/|A|)\ge p^{\alpha }/4$. This supplies a positive exponent depending only on $\alpha ,\beta$. The finitely many smaller primes are covered by the strict inequality $|A+bA|>|A|$ for every nonzero $b$ when $1<|A|<p$.

For $S\subseteq {\mathbb{F}}_p$ with $|S|\ge 2$, put $R=(S-S)/(S-S)$ using nonzero denominators. If $R$ is proper, some $r\in R+1$ lies outside $R$, since invariance under addition of one would give all of the prime field. The map $(x,y)\mapsto x+ry$ on $S^2$ is injective. Clearing the denominator in a representation of $r-1$ puts a dilate of its image inside $3SS-3SS$. If $R$ is the full field, averaging the collision count of $S+rS$ gives some $r$ with $|S+rS|\ge \frac{1}{2}\min (p,|S{|}^2)$. Clearing its denominator puts a dilate inside $2SS-2SS\subseteq 3SS-3SS$. Here $SS$ is a product set, and the prefactors denote repeated sumsets, not scalar multiplication.

For finite $U,V,W\subseteq \mathbb{C}$ with $U\ne \{0\}$ and $W\ne \{0\}$,In particular, addition and multiplication cannot both expand a nontrivial finite set of complex numbers only slightly.