Multidimensional Szemerédi Theorem Bounds

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a fixed dimension $d\ge1$ and a finite pattern $F\subset\mathbb Z^d$ with at least two points, a nontrivial homothetic copy is $x+rF=\{x+rv:v\in F\}$, where $x\in\mathbb Z^d$ and $r\in\mathbb N$. Put \[ r_F(N)=\max\{|A|:A\subset\{1,\ldots,N\}^d,\ A\text{ contains no nontrivial copy of }F\}. \] Define $\log_1 t=\log t$ and $\log_{m+1}t=\log(\log_m t)$ for sufficiently large $t$. A density threshold of finite logarithmic type has the form $C/\log_m M$ with a fixed integer $m\ge1$, a constant $C>0$, and $M$ the cardinality of the ambient finite structure. This is the minimum quantitative scale proposed in the source. For a prime $p$, let $\mathbb F_p$ be the field with $p$ elements and $G_n=(\mathbb F_p)^n$. A square in $G_n^2$ is \[ (x,y),\ (x+h,y),\ (x,y+h),\ (x+h,y+h),\qquad h\in G_n\setminus\{0\}. \] A three-dimensional corner in $G_n^3$ is \[ (x,y,z),\ (x+h,y,z),\ (x,y+h,z),\ (x,y,z+h),\qquad h\ne0. \]

Statement recorded in the supplied source.

Find explicit quantitative density bounds of finite logarithmic type for instances of the multidimensional Szemerédi theorem. In the integer-grid formulation, the target is: for each fixed $d,F$, find $m,C$ such that \[ r_F(N)\le\frac{C N^d}{\log_m(N^d)} \] for all sufficiently large $N$. The source specifically asks for such bounds for square-free subsets of $G_n^2$ and three-dimensional-corner-free subsets of $G_n^3$, with $m,C$ independent of $n$ (and allowed to depend on the fixed prime $p$ and the configuration). Here the ambient sizes are $p^{2n}$ and $p^{3n}$ respectively. The request concerns general instances of the theorem as well as these particular configurations. [S1][S2]

Short English statement

Obtain density thresholds controlled by a fixed finite number of logarithms for multidimensional arithmetic configurations, including squares and three-dimensional corners over finite fields, and fixed homothetic patterns in integer grids.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 108 (GREEN-020). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green, 100 Open Problems, author manuscript with December 2025 updates, Problem 20 and its comments, pp. 12--13. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.