Rogers' mean value theorem for S-arithmetic Siegel transforms and applications to the geometry of numbers

Values of inhomogeneous forms at S ‐integral points

“…Remark We note that if $$q=1$$, then Theorem 2.1 follows from [16]. Hence throughout this paper, let us assume that $$1\ne q\in {\mathbb {N}}_S$$.…”

“…The following theorem is found in [17] (Lemma 3.10) for SL 𝑑 (ℤ 𝑆 ) ⧵ SL 𝑑 (ℚ 𝑆 ) and [16] (Proposition 3.2) for UL 𝑑 (ℤ 𝑆 ) ⧵ UL 𝑑 (ℚ 𝑆 ).…”

“…The following theorem is found in [17] (Lemma 3.10) for $$\mathrm{SL}_d({\mathbb {Z}}_S)\setminus \mathrm{SL}_d({\mathbb {Q}}_S)$$ and [16] (Proposition 3.2) for $$\mathrm{UL}_d({\mathbb {Z}}_S)\setminus \mathrm{UL}_d({\mathbb {Q}}_S)$$. Theorem For $$1\leqslant r < d$$, $$\begin{equation*} \int _{\mathrm{UL}_d({\mathbb {Z}}_S)\setminus \mathrm{UL}_d({\mathbb {Q}}_S)} \alpha ^r d\mu _S^{(d)} < \infty \quad \text{and}\quad \int _{\mathrm{SL}_d({\mathbb {Z}}_S)\setminus \mathrm{SL}_d({\mathbb {Q}}_S)} \alpha ^r d\mu _S^{(d)} < \infty .$$…”

“…The case of forms of rank 3 and 4 was studied by Han in [18]. Effective results for generic (homogeneous) forms in the S ‐arithmetic setting were obtained by Han in [16], who also established an S ‐arithmetic version of Rogers' mean value formula. Finally, we mention that the question of values of rational quadratic forms with congruence conditions has also received attention recently, see for instance [4].…”

“…Proof of Theorem 5.1. We will follow the strategy of the proof of Theorem 2.10 in[16], which is based on the Borel-Cantelli lemma. We first fix an arbitrary compact set  in UL 𝑑 (ℚ 𝑆 ).Let (𝛿 𝑝 ) 𝑝∈𝑆 be an 𝑆-tuple of positive real numbers.…”