Gorodnik, Alexander; Nevo, Amos Lifting, restricting and sifting integral points on affine homogeneous varieties. (English) Zbl 1307.11078 Compos. Math. 148, No. 6, 1695-1716 (2012). Authors’ abstract: “In [J. Reine Angew. Math. 663, 127–176 (2012; Zbl 1248.37011)], an effective solution of the lattice point counting problem in general domains in semi simple \(S\)-algebraic groups and affine symmetric varieties. The method relies on the mean ergodic theorem for the action of \(G\) on \(G/\Gamma\), and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [the second author and P. Sarnak, Acta Math. 205, No. 2, 361–402 (2010; Zbl 1233.11102)] and use them to establish several useful consequences of this property, including: (1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties; (2) effective upper bounds on the number of integral points on general sub varieties of semi simple group varieties; (3) effective lower bounds on the number of almost prime points on symmetric varieties; (4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.”The results have also been illustrated through examples. Reviewer: Ranjeet Sehmi (Chandigarh) Cited in 5 Documents MSC: 11G35 Varieties over global fields 11N36 Applications of sieve methods 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values 11D79 Congruences in many variables Keywords:lifting; restricting; sifting; homogeneous varieties; effective bounds Citations:Zbl 1248.37011; Zbl 1233.11102 PDFBibTeX XMLCite \textit{A. Gorodnik} and \textit{A. Nevo}, Compos. Math. 148, No. 6, 1695--1716 (2012; Zbl 1307.11078) Full Text: DOI arXiv References: [2] doi:10.1353/ajm.0.0034 · Zbl 1231.14041 [4] doi:10.1016/0022-4049(73)90014-5 · Zbl 0277.14005 [6] doi:10.1215/S0012-7094-93-07108-6 · Zbl 0798.11025 [8] doi:10.1215/S0012-7094-93-07107-4 · Zbl 0798.11024 [12] doi:10.1080/00927877508822057 · Zbl 0315.12001 [15] doi:10.1215/S0012-7094-02-11314-3 · Zbl 1011.22007 [16] doi:10.1112/blms/11.1.55 · Zbl 0425.12003 [17] doi:10.1112/S0010437X0800376X · Zbl 1190.11036 [18] doi:10.1007/s11511-010-0057-4 · Zbl 1233.11102 [19] doi:10.1007/s00222-002-0253-8 · Zbl 1025.11012 [20] doi:10.1007/BF01243900 · Zbl 0774.11021 [21] doi:10.1007/BF01207477 · Zbl 0641.12004 [22] doi:10.1215/S0012-7094-06-13236-2 · Zbl 1098.14013 [24] doi:10.1007/978-3-0346-0129-0 [25] doi:10.1215/S0012-7094-07-13626-3 · Zbl 1117.22006 [26] doi:10.1007/s00222-009-0225-3 · Zbl 1239.11103 [27] doi:10.1007/s11856-010-0069-y · Zbl 1230.11045 [29] doi:10.2307/1970210 · Zbl 0107.14804 [31] doi:10.1007/978-94-011-0141-7_8 [32] doi:10.2307/2372655 · Zbl 0058.27202 [33] doi:10.1007/978-3-0348-7662-9_5 [35] doi:10.1007/978-3-540-36364-4_2 · Zbl 1152.11027 [36] doi:10.2307/3062125 · Zbl 1039.11044 [39] doi:10.2140/ant.2008.2.595 · Zbl 1194.14067 [41] doi:10.1007/s00039-006-0583-6 · Zbl 1112.37001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.