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

Han, Jiyoung

doi:10.48550/arxiv.1910.01824

Cited by 5 publications

(6 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is easy to translate Theorem 1.2 into the classical language, and unsurprisingly, certain recently proved variants of the Rogers formula (e.g., [1,14]) follow immediately this way. Let us also demonstrate a quick example involving a number field other than ℚ.…”

Section: 2mentioning

confidence: 98%

Adelic Rogers integral formula

Kim

2023

Journal of London Math Soc

View full text Add to dashboard Cite

show abstract

Section: 2mentioning

confidence: 98%

Adelic Rogers integral formula

Kim

2023

Journal of London Math Soc

View full text Add to dashboard Cite

show abstract

“…Proof of Theorem 5.1. We will follow the strategy of the proof of Theorem 2.10 in [17], which is based on the Borel-Cantelli lemma. We first fix an arbitrary compact set K in UL d (Q S ).…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…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 [17], 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 [5].…”

Section: Introductionmentioning

confidence: 99%

Values of Inhomogeneous Forms at S-integral points

Ghosh¹,

Han²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We prove effective versions of Oppenheim's conjecture for generic inhomogeneous forms in the S-arithmetic setting. We prove an effective result for fixed rational shifts and generic forms and we also prove a result where both the quadratic form and the shift are allowed to vary. In order to do so, we prove analogues of Rogers' moment formulae for S-arithmetic congruence quotients as well as for the space of affine lattices. We believe the latter results to be of independent interest.

show abstract

“…More recently, there are versions for the rational points on Grassmannians ( [14]), translation surfaces ( [43], see also related [2]), and for cut-and-project sets ( [27]), the latter two proved by ergodic theoretic methods. There are also extensions to S-unit lattices ( [11]), to ASL(n, R) ( [8], [1]), and to sums over translates of Z n ( [10], [1]), which requires one to consider the quotient of SL(n, R) with respect to a congruence subgroup.…”

Section: Introductionmentioning

confidence: 99%