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The hyperbolic lattice point count in infinite volume with applications to sieves. (English) Zbl 1223.11113

J. Bourgain, A. Gamburd and P. Sarnak [Sieving and expanders, C. R., Math., Acad. Sci. Paris 343, No. 3, 155–159 (2006; Zbl 1217.11081), Invent. Math. 179, No. 3, 559–644 (2010; Zbl 1239.11103)] and P. Sarnak [Astérisque 322, 225–240 (2008; Zbl 1223.11112)] proved the following: Let \(\Gamma\subset \text{SL}_2(\mathbb Z)\) be any non-elementary subgroup, let \(q \in \mathbb Z^2\) be nonzero, and let \(O=q \cdot G\) be a \(G\)-orbit. Let \(f: \mathbb Z^2 \to \mathbb Z\) be any polynomial. Then there exists an \(R<\infty\), depending on all of the above data, such that there are infinitely many points in the set \(f(O)\) having at most \(R\) prime factors.
In the paper under review, the author proves a special version of the above theorem along with some applications to representing almost primes in thin sets. More specifically, consider the problem of primes in the sum of two squares, \(f(c,d)=c^2+d^2\), but restrict \((c,d)\) to the orbit \(\O=(0,1) G\), where \(G\) is an infinite-index non-elementary finitely-generated subgroup of \(\text{SL}_2(\mathbb Z)\). Assume that the Riemann surface \(G \setminus H\) has a cusp at infinity. The author shows that the set of values \(f(\theta)\) contains infinitely many integers having at most \(R\) prime factors for any \(R>4/(\delta-\theta)\), where \(\theta > 1/2\) is the spectral gap and \(\delta<1\) is the Hausdorff dimension of the limit set of \(G\). If \(\delta>149/150\), then we can take \(\theta=5/6\), giving \(R=25\). The limit of this method is \(R=9\) for \(\delta-\theta>4/9\). This is the same number of prime factors as attained in Brun’s original attack on the twin prime conjecture.
In the infinite volume case, one does not have the the explicit Eisenstein series and spectral expansion theorem (as in the finite volume case) to facilitate computations over the continuous spectrum. So the author uses abstract spectral theory directly instead.

MSC:

11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11P21 Lattice points in specified regions
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11N36 Applications of sieve methods
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References:

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