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<p>This paper generalizes the result of Sarnak and Ubis [9] about non-concentration of primes in horocycle orbits on PSL<sub>2</sub>(ℤ)PSL<sub>2</sub>(ℝ) to any lattice in PSL<sub>2</sub>(ℝ). The proof combines the asymptotic result of Strömbergsson [11] and Venkatesh’s method [12] with the approach of Sarnak and Ubis of approximating horocycle pieces with periodic horocycles. The key step is to establish a dichotomy between {<i>ξh</i>(<i>t</i>), <i>t</i> ∈ [0, <i>T</i>]} having good equidistribution in ΓPSL<sub>2</sub> (ℝ) and it being approximable by closed horocycle pieces with small period. In a follow-up paper, a similar approach will be used to show equidistribution of <i>ξh</i>(<i>n</i><sup>1+<i>γ</i></sup>) for small <i>γ</i> &gt; 0, generalizing Venkatesh’s result [12] to non-compact Γ.</p>

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Non-concentration of primes in Γ\PSL2(ℝ)

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  • 发表日期：

    2025年3月27日

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  • Lauritz Streck

摘要

This paper generalizes the result of Sarnak and Ubis [9] about non-concentration of primes in horocycle orbits on PSL2(ℤ)PSL2(ℝ) to any lattice in PSL2(ℝ). The proof combines the asymptotic result of Strömbergsson [11] and Venkatesh’s method [12] with the approach of Sarnak and Ubis of approximating horocycle pieces with periodic horocycles. The key step is to establish a dichotomy between {ξh(t), t ∈ [0, T]} having good equidistribution in ΓPSL2 (ℝ) and it being approximable by closed horocycle pieces with small period. In a follow-up paper, a similar approach will be used to show equidistribution of ξh(n1+γ) for small γ > 0, generalizing Venkatesh’s result [12] to non-compact Γ.