U sklopu Seminara za geometriju u ponedjeljak, 23.06.2025. s početkom u 16 sati u predavaonici 108  održat će se predavanje:

Ji Zeng (Alfréd Rényi Institute of Mathematics), "Evasive sets, twisted varieties, and container-clique trees".

Sažetak:

A set $S$ in affine space $\mathbb{F}_q^n$ is said to be $(d,k,c)$-evasive if the intersection between $S$ and any variety in $\mathbb{F}_q^n$, of dimension $k$ and degree at most $d$, has cardinality at most $c$. By a simple averaging argument, we must have $|S| \leq O\left(q^{n-k}\right)$ as $q\to\infty$. We exhibit the existence of such evasive sets of size $\Omega\left(q^{n-k}\right)$ for values of $c$ much smaller than previously known constructions. We also establish an upper bound $2^{O(q^{n-k})}$ for the total number of such evasive sets which is asymptotically tight in the exponent.

The existence result is based on our study of twisted varieties. A variety $V$ in projective space $\mathbb{P}^n$ is said to be $d$-twisted if the intersection between $V$ and any other variety in $\mathbb{P}^n$, of codimension $\dim(V)$ and degree at most $d$, has dimension zero. We prove an upper bound on the smallest possible degree for twisted varieties which is best possible for complete intersections.

The enumeration result includes a new  trick for the container method called container-clique trees, which we believe is of independent interest. To illustrate the potential of this trick, we give a simpler proof of a result by Chen—Liu—Nie—Zeng that characterizes the maximum size of a collinear-triple-free subset in a random sampling of $ \mathbb{F}_q^2$ up to polylogarithmic factors. Joint work with Jeck Lim and Jiaxi Nie.