Abstract
Abstract
Let
$\mathbb {F}_q^d$
denote the d-dimensional vector space over the finite field
$\mathbb {F}_q$
with q elements. Define for
$\alpha = (\alpha _1, \dots , \alpha _d) \in \mathbb {F}_q^d$
. Let
$k\in \mathbb {N}$
, A be a nonempty subset of
$\{(i, j): 1 \leq i < j \leq k + 1\}$
and
$r\in (\mathbb {F}_q)^2\setminus {0}$
, where
$(\mathbb {F}_q)^2=\{a^2:a\in \mathbb {F}_q\}$
. If
$E\subset \mathbb {F}_q^d$
, our main result demonstrates that when the size of the set E satisfies
$|E| \geq C_k q^{d/2}$
, where
$C_k$
is a constant depending solely on k, it is possible to find two
$(k+1)$
-tuples in E such that one of them is dilated by r with respect to the other, but only along
$|A|$
edges. To be more precise, we establish the existence of
$(x_1, \dots , x_{k+1}) \in E^{k+1}$
and
$(y_1, \dots , y_{k+1}) \in E^{k+1}$
such that, for
$(i, j) \in A$
, we have
$\lVert y_i - y_j \rVert = r \lVert x_i - x_j \rVert $
, with the conditions that
$x_i \neq x_j$
and
$y_i \neq y_j$
for
$1 \leq i < j \leq k + 1$
, provided that
$|E| \geq C_k q^{d/2}$
and
$r\in (\mathbb {F}_q)^2\setminus \{0\}$
. We provide two distinct proofs of this result. The first uses the technique of group actions, a powerful method for addressing such problems, while the second is based on elementary combinatorial reasoning. Additionally, we establish that in dimension 2, the threshold
$d/2$
is sharp when
$q \equiv 3 \pmod 4$
. As a corollary of the main result, by varying the underlying set A, we determine thresholds for the existence of dilated k-cycles, k-paths and k-stars (where
$k \geq 3$
) with a dilation ratio of
$r\in (\mathbb {F}_q)^2\setminus \{0\}$
. These results improve and generalise the findings of Xie and Ge [‘Some results on similar configurations in subsets of
$\mathbb {F}_q^d$
’, Finite Fields Appl.91 (2023), Article no. 102252, 20 pages].
Publisher
Cambridge University Press (CUP)