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  2019, 15: 277-327. doi: 10.3934/jmd.2019022

Almost-prime times in horospherical flows on the space of lattices

Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT 06520-8283, USA

Received  June 06, 2018 Revised  April 20, 2019 Published  November 2019

An integer is called almost-prime if it has fewer than a fixed number of prime factors. In this paper, we study the asymptotic distribution of almost-prime entries in horospherical flows on $ \Gamma\backslash {{\rm{SL}}}_n(\mathbb{R}) $, where $ \Gamma $ is either $ {{\rm{SL}}}_n(\mathbb{Z}) $ or a cocompact lattice. In the cocompact case, we obtain a result that implies density for almost-primes in horospherical flows where the number of prime factors is independent of basepoint, and in the space of lattices we show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. Along the way we give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, as well as an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows.

Citation: Taylor McAdam. Almost-prime times in horospherical flows on the space of lattices. Journal of Modern Dynamics, 2019, 15: 277-327. doi: 10.3934/jmd.2019022
References:
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A018804, On-line encyclopedia of integer sequences, https://oeis.org/A018804 (accessed Feb. 2, 2018).

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O. Bordellès, Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Seq., 10 (2007), Article 07.9.2, 13pp.

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O. Bordellès, A note on the average order of the gcd-sum function, J. Integer Seq., 10 (2007), Article 07.3.3, 4pp.

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J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (Israel Math. Seminar), Lecture Notes in Math., 1318, Springer Berlin, 1988,204–223. doi: 10.1007/BFb0081742.

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J. BourgainP. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, Developments in Mathematics, 28 (2013), 67-83.  doi: 10.1007/978-1-4614-4075-8_5.

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K. A. Broughan, The gcd-sum function, J. Integer Seq., 4 (2001), Article 01.2.2, 19pp.

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K. A. Broughan, The average order of the Dirichlet series of the gcd-sum function, J. Integer Seq., 10 (2007), Article 07.4.2, 6pp.

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J. A. Brudnyĭ and M. I. Ganzburg, On an extremal problem for polynomials in $n$ variables, Math. USSR Izv., 7 (1973), 344-355. 

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M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.

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E. Cesàro, Étude moyenne di plus grand commun diviseur de deux nombres, Annali di Matematica Pura ed Applicata (1867-1897), 13 (1885), 235-250. 

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J. Chidambaraswamy and R. Sitaramachandra Rao, Asymptotic results for a class of arithmetical functions, Monatsh. Math., 99 (1985), 19-27.  doi: 10.1007/BF01300735.

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K. DabbsM. Kelly and H. Li, Effective equidistribution of translates of maximal horospherical measures in the space of lattices, J. Mod. Dyn., 10 (2016), 229-254.  doi: 10.3934/jmd.2016.10.229.

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M. EinsiedlerG. A. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.

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L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

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H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, CT, 1972), Lecture Notes in Math., 318, Springer, Berlin, 1973, 95–115.

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A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35.  doi: 10.3934/jmd.2007.1.1.

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A. GorodnikF. Maucourant and H. Oh, Manin's and Peyre's conjectures on rational points and adelic mixing, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 383-435.  doi: 10.24033/asens.2071.

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B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[32] H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974. 
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P. Haukkanen, On a gcd-sum function, Aequationes Math., 76 (2008), 168-178.  doi: 10.1007/s00010-007-2923-5.

[34]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542.  doi: 10.1215/S0012-7094-36-00246-6.

[35]

H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloquium Publications, 53, Amer. Math. Society, Providence, RI, 2004. doi: 10.1090/coll/053.

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A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Publications mathématiques de l'IHÉS, 79 (1994), 131–156.

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D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2,171, Adv. Math. Soc., Providence, RI, 1996,141–172. doi: 10.1090/trans2/171/11.

[38]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.

[39]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[40]

D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, Boston, 2012,385–396. doi: 10.1007/978-1-4614-1260-1_18.

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A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, 140, Birkhäuser Boston, Inc., 1996. doi: 10.1007/978-1-4757-2453-0.

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M. Lee and H. Oh, Effective equidistribution of closed horocycles for geometrically finite surfaces, preprint, arXiv: 1202.0848, (2012).

[43]

E. Lindenstrauss, G. A. Margulis, A. Mohammadi and N. Shah, Quantitative behavior of unipotent flows and an effective avoidance principle, preprint, arXiv: 1904.00290, (2019).

[44]

J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J., 164 (2015), 1353-1399.  doi: 10.1215/00127094-2916213.

[45]

G. A. Margulis, On some Aspects of the Theory of Anosov Systems, Ph.D. thesis, Lomonosov Moscow State University (1970) (in Russian); English transl.: Springer Monographs in Mathematics, Springer, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.

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G. A. Margulis, On the action of unipotent groups in a lattice space, Mat. Sb. (N.S.), 86 (1971), 552-556. 

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G. A. Margulis, On the action of unipotent groups in the space of lattices, in Lie Groups and Their Representations (Proc. Summer School, Bolyai, János Math. Soc., Budapest, 1971), Wiley, New York, 1975,365–370.

[48]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogènes, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 249-253. 

[49]

G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Symposium in honor of Atle Selberg, Oslo, 1987), Academic Press, Boston, 1989, 377–398.

[50]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.  doi: 10.1007/BF01231565.

[51]

A. Nevo and P. Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math., 205 (2010), 361-402.  doi: 10.1007/s11511-010-0057-4.

[52]

H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Math. Soc. France, 126 (1998), 355-380.  doi: 10.24033/bsmf.2329.

[53]

H. Oh, Uniform pointwise bounds for decay of matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192.  doi: 10.1215/S0012-7094-02-11314-3.

[54]

R. Peckner, Möbius disjointness for homogeneous dynamics, Duke Math. J., 167 (2018), 2745-2792.  doi: 10.1215/00127094-2018-0026.

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S. S. Pillai, On an arithmetic function, J. of the Annamalai Univ., 2 (1933), 243-248. 

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M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, New York-Heidelberg, 1972.

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M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.

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M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal., 4 (1994), 236-257.  doi: 10.1007/BF01895839.

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E. J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow, 13 (1936), 93-95. 

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P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739.  doi: 10.1002/cpa.3160340602.

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show all references

References:
[1]

A018804, On-line encyclopedia of integer sequences, https://oeis.org/A018804 (accessed Feb. 2, 2018).

[2] M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, Cambridge University Press, 2000.  doi: 10.1017/CBO9780511758898.
[3]

Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble), 62 (2012), 1889-1942.  doi: 10.5802/aif.2738.

[4]

O. Bordellès, Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Seq., 10 (2007), Article 07.9.2, 13pp.

[5]

O. Bordellès, A note on the average order of the gcd-sum function, J. Integer Seq., 10 (2007), Article 07.3.3, 4pp.

[6]

J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (Israel Math. Seminar), Lecture Notes in Math., 1318, Springer Berlin, 1988,204–223. doi: 10.1007/BFb0081742.

[7]

J. BourgainP. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, Developments in Mathematics, 28 (2013), 67-83.  doi: 10.1007/978-1-4614-4075-8_5.

[8]

K. A. Broughan, The gcd-sum function, J. Integer Seq., 4 (2001), Article 01.2.2, 19pp.

[9]

K. A. Broughan, The average order of the Dirichlet series of the gcd-sum function, J. Integer Seq., 10 (2007), Article 07.4.2, 6pp.

[10]

J. A. Brudnyĭ and M. I. Ganzburg, On an extremal problem for polynomials in $n$ variables, Math. USSR Izv., 7 (1973), 344-355. 

[11]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.

[12]

E. Cesàro, Étude moyenne di plus grand commun diviseur de deux nombres, Annali di Matematica Pura ed Applicata (1867-1897), 13 (1885), 235-250. 

[13]

J. Chidambaraswamy and R. Sitaramachandra Rao, Asymptotic results for a class of arithmetical functions, Monatsh. Math., 99 (1985), 19-27.  doi: 10.1007/BF01300735.

[14]

K. DabbsM. Kelly and H. Li, Effective equidistribution of translates of maximal horospherical measures in the space of lattices, J. Mod. Dyn., 10 (2016), 229-254.  doi: 10.3934/jmd.2016.10.229.

[15]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138.  doi: 10.1007/BF01578067.

[16]

S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385.  doi: 10.1007/BF01389173.

[17]

S. G. Dani, On orbits of unipotent flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 4 (1984), 25-34.  doi: 10.1017/S0143385700002248.

[18]

S. G. Dani, On orbits of unipotent flows on homogeneous spaces II, Ergodic Theory Dynam. Systems, 6 (1986), 167-182.  doi: 10.1017/S0143385700003382.

[19]

S. G. Dani, Orbits of horospherical flows, Duke Math. J., 53 (1986), 177-188.  doi: 10.1215/S0012-7094-86-05312-3.

[20]

S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.  doi: 10.1007/BF02872005.

[21]

H. Davenport, Multiplicative Number Theory, Graduate Texts in Mathematics, 74, Springer-Verlag, 1980.

[22]

L. E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Primality, Chelsea Publishing Co., 1966.

[23]

M. Einsiedler, G. A. Margulis, A. Mohammadi and A. Venkatesh, Effective equidistribution and property tau, Effective Equidistribution and Property $ (\tau)$, (2019), 1–77. doi: 10.1090/jams/930.

[24]

M. EinsiedlerG. A. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.

[25]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Springer-Verlag London, 2011. doi: 10.1007/978-0-85729-021-2.

[26]

R. Ellis and W. Perrizo, Unique ergodicity of flows on homogeneous spaces, Israel J. Math., 29 (1978), 276-284.  doi: 10.1007/BF02762015.

[27]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

[28]

H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, CT, 1972), Lecture Notes in Math., 318, Springer, Berlin, 1973, 95–115.

[29]

A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35.  doi: 10.3934/jmd.2007.1.1.

[30]

A. GorodnikF. Maucourant and H. Oh, Manin's and Peyre's conjectures on rational points and adelic mixing, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 383-435.  doi: 10.24033/asens.2071.

[31]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[32] H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974. 
[33]

P. Haukkanen, On a gcd-sum function, Aequationes Math., 76 (2008), 168-178.  doi: 10.1007/s00010-007-2923-5.

[34]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542.  doi: 10.1215/S0012-7094-36-00246-6.

[35]

H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloquium Publications, 53, Amer. Math. Society, Providence, RI, 2004. doi: 10.1090/coll/053.

[36]

A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Publications mathématiques de l'IHÉS, 79 (1994), 131–156.

[37]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2,171, Adv. Math. Soc., Providence, RI, 1996,141–172. doi: 10.1090/trans2/171/11.

[38]

D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2), 148 (1998), 339-360.  doi: 10.2307/120997.

[39]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[40]

D. Y. Kleinbock and G. A. Margulis, On effective equidistribution of expanding translates of certain orbits in the space of lattices, in Number Theory, Analysis and Geometry, Springer, Boston, 2012,385–396. doi: 10.1007/978-1-4614-1260-1_18.

[41]

A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, 140, Birkhäuser Boston, Inc., 1996. doi: 10.1007/978-1-4757-2453-0.

[42]

M. Lee and H. Oh, Effective equidistribution of closed horocycles for geometrically finite surfaces, preprint, arXiv: 1202.0848, (2012).

[43]

E. Lindenstrauss, G. A. Margulis, A. Mohammadi and N. Shah, Quantitative behavior of unipotent flows and an effective avoidance principle, preprint, arXiv: 1904.00290, (2019).

[44]

J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J., 164 (2015), 1353-1399.  doi: 10.1215/00127094-2916213.

[45]

G. A. Margulis, On some Aspects of the Theory of Anosov Systems, Ph.D. thesis, Lomonosov Moscow State University (1970) (in Russian); English transl.: Springer Monographs in Mathematics, Springer, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.

[46]

G. A. Margulis, On the action of unipotent groups in a lattice space, Mat. Sb. (N.S.), 86 (1971), 552-556. 

[47]

G. A. Margulis, On the action of unipotent groups in the space of lattices, in Lie Groups and Their Representations (Proc. Summer School, Bolyai, János Math. Soc., Budapest, 1971), Wiley, New York, 1975,365–370.

[48]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogènes, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 249-253. 

[49]

G. A. Margulis, Discrete subgroups and ergodic theory, in Number Theory, Trace Formulas and Discrete Groups (Symposium in honor of Atle Selberg, Oslo, 1987), Academic Press, Boston, 1989, 377–398.

[50]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.  doi: 10.1007/BF01231565.

[51]

A. Nevo and P. Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math., 205 (2010), 361-402.  doi: 10.1007/s11511-010-0057-4.

[52]

H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients, Bull. Math. Soc. France, 126 (1998), 355-380.  doi: 10.24033/bsmf.2329.

[53]

H. Oh, Uniform pointwise bounds for decay of matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113 (2002), 133-192.  doi: 10.1215/S0012-7094-02-11314-3.

[54]

R. Peckner, Möbius disjointness for homogeneous dynamics, Duke Math. J., 167 (2018), 2745-2792.  doi: 10.1215/00127094-2018-0026.

[55]

S. S. Pillai, On an arithmetic function, J. of the Annamalai Univ., 2 (1933), 243-248. 

[56]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, New York-Heidelberg, 1972.

[57]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.

[58]

M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal., 4 (1994), 236-257.  doi: 10.1007/BF01895839.

[59]

E. J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow, 13 (1936), 93-95. 

[60]

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Figure 1.  The symmetric difference between $ B_T $ and $ B_T - (H,\ldots,H) $
Figure 2.  The measure of the set where $ |{\bf s_1-s_2| }<\epsilon $ has measure bounded by $ H^d\epsilon^d $ in $ B_H\times B_H $ (shown here for one dimensional $ U $)
Figure 3.  The area shaded in solid gray indicates the region over which we are integrating in the definition of $ S_{\rm approx} $, whereas the area shaded with diagonal lines represents the region over which we are integrating in our estimate of $ S_{\rm approx} $ given in (43). The difference between the two integrals can be bounded by the number of $ \delta $-cubes intersecting the boundary of $ B_T $ multiplied by the supremum of $ f $
Figure 4.  In $ S_K(A) $ we are summing over the integer points in $ \tilde B_T $ such that $ K|k_1\cdots k_2 $ (filled in black). We may do this by summing over shifted grids based at each of the points in the first box $ \tilde B_K $ (filled in gray). However, this introduces an error determined by $ \mathscr{S}_{{\infty},{0}}(f) $ and the number of points in each of these shifted grids falling outside $ B_T $ (filled in white). The number of such points can be bounded by $ T^{d-1}K^{1-d} $, as we have seen before
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