The positive-density patterns conjecture for affine buildings

Let $\Delta$ be an irreducible affine building with uniform thickness $2\leq q<\infty$. Let $X\subseteq V_P$ have positive upper density $d^*(X)>0$. For each $k>0$, there exists $K=K(X,k)>0$ such that whenever

$$\lambda_1,\ldots,\lambda_k\in Q\cap P^+$$

satisfy $\lambda_k\geq\cdots\geq\lambda_2\geq\lambda_1\geq K\rho$, there exists a subset $\{v_0,v_1,\ldots,v_k\}\subseteq X$ such that

$$\boldsymbol{d}(v_0,v_j)=\lambda_j\quad\text{for all }1\leq j\leq k$$

and

$$\boldsymbol{d}(o,v_0)=\boldsymbol{d}(o,v_1)=\cdots=\boldsymbol{d}(o,v_k).$$

Positive-density patterns conjecture. Every positive-density subset of the vertex set of an irreducible affine building contains these arbitrarily large prescribed configurations with pairwise vector distances from one distinguished vertex and all vertices lying at the same distance from the base vertex. The preceding theorem establishes the analogous result for $(-1)$-type affine buildings, while the remark explains that its current formulation fails for non-$(-1)$-type buildings; the conjecture proposes that the stated formulation remains valid when the prescribed distances lie in $Q\cap P^+$.