Hardy-Littlewood varieties and semisimple groups

Mikhail Borovoi*, Zeév Rudnick

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for which the Hardy-Littlewood circle method is applicable are strongly Hardy-Littlewood. We prove that certain affine homogeneous spaces of semisimple groups are strongly Hardy-Littlewood varieties. Moreover, we prove that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties for with the asymptotic density of integer points can be computed in terms of a product of local densities.

Original languageEnglish
Pages (from-to)37-66
Number of pages30
JournalInventiones Mathematicae
Issue number1
StatePublished - Dec 1995


Dive into the research topics of 'Hardy-Littlewood varieties and semisimple groups'. Together they form a unique fingerprint.

Cite this