Three-variable expanding polynomials and higher-dimensional distinct distances (Q2322508)

Let \(\mathbb{F}\) be a field of characterstic \(p>0\), or, if its the characteristic is 0, set \(p=\infty\). A polynomial \(f \in \mathbb{F}[x_1, \dots, x_k]\) is called an \textit{expander} if there are \(\alpha > 1\), \(\beta > 0\) such that foar all sets \(A_1, \dots, A_k \subset \mathbb{F}\) of size \(N \ll p^\beta\) we have \(\vert f(A_1\times \dots \times A_k\vert \gg N^\alpha\). Given \(P \subset \mathbb{F}^d\), the \textit{distance set} is defined as \[ \Delta(P):= \left\{\sum_{i=1}^d (x_i-y_i)^2 : (x_1, \dots, x_d), (y_1, \dots, y_d) \in P\right\}.\] \par Here are the main results.\par Theorem 1. Let \(f\in \mathbb{F}[x, y, z]\) be a quadratic polynomial that depends on each variable and is not of the form \(g(h(x) + k(y) + l(z))\). Let \(A, B, C \subset F\) with \(\vert A\vert = \vert B\vert = \vert C\vert = N\). Then \[\vert f(A\times B \times C)\vert \gg \min\left\{N^{3/2}, p\right\}.\]\par Theorem 2. For \(A\subset \mathbb{F}\) with \(\vert A\vert \ll p^{5/8}\), we have \(\vert A + A^2\vert \gg \vert A\vert ^{6/5}\). \par Theorem 3. For \(A\subset \mathbb{F}\) with \(\vert A\vert \ll p^{5/8}\), \(\max \left\{\vert A + A\vert, \vert A^2 + A^2\vert \right\} \gg \vert A\vert ^{6/5}\). \par Theorem 4. For \(A\subset \mathbb{F}\), we have \(\vert \Delta(A^d)\vert \gg_d \min \left\{\vert A\vert ^{2-\frac{1}{2^{d-1}}}, p \right\}\).\par Theorem 5. For \(A\subset \mathbb{R}\) and \(d\geq 2\), we have \(\vert \Delta(A^d)\vert \gg \frac{\vert A\vert ^2}{\log^{1/2^{d-2}}\vert A\vert }\).\par Two more complex theorems deal with higher dimensions.