# Research

"Every simple thought process adds something permanent and substantial to our soul." - B. Riemann

(German original: Mit jedem einfachen Denkakt tritt etwas Bleibendes, Substantielles in unsere Seele ein.)

Collaborators (# of collaborations):

Ch. Aistleitner (4)

S. Baker (1)

S. Chow (5)

Ch. Elsholtz (2)

C. Elsner (1)

T. Lachmann (3)

Ch. Lutsko (3)

M. Munsch (1)

F. Pazuki (1)

Z. Rudnick (2)

M. Radziwiłł (1)

D. Schindler (1)

A. Sourmelidis (1)

R. Srivastava (2)

M. Technau (2)

R. Tichy (1)

M. Widmer (2)

A. Walker (1)

N. Yesha (2)

Selected manuscripts:

S. Chow, and N. Technau: Smooth discrepancy and Littlewood's conjecture.

(Here is the arXiv version.)S. Chow, and N. Technau: Littlewood and Duffin--Schaeffer type problems in diophantine approximation,

Mem. Amer. Math. Soc. (2024), DOI: https://doi.org/10.1090/memo/1475.

(Here is the arXiv version.)Ch. Lutsko, and N. Technau: Full Poissonian Local Statistics of Slowly Growing Sequences.

to appear in Compositio. Math.

(Here is the arXiv version.)N. Technau, and N. Yesha: On the correlations of $n^\alpha$ mod 1.

J. Eur. Math. Soc. 25 (2023), no. 10, pp. 4123–4154, DOI: 10.4171/JEMS/1281.

(Here is the arXiv version.)Z. Rudnick, and N. Technau: The metric theory of the pair correlation function for small non-integer powers.

J. London Math. Soc., DOI: 10.1112/jlms.12647.

(Here is the arXiv version.)R. Srivastava, and N. Technau: Density of Rational Points Near Flat/Rough Hypersurfaces.

(Here is the arXiv version.)D. Schindler, R. Srivastava, and N. Technau: Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals.

(Here is the arXiv version.)M. Radziwiłł, and N. Technau: The gap distribution of $\sqrt{n} mod 1$ and the circle method.

(Here is the arXiv version.)

General Audience Articles and Exposition about my work:

Scientific American: Chris Lutsko, wrote an elegant exposition about our joint papers and our work with Athanasios Sourmelidis. We studied the pseudo-randomness of monomial sequences with tools from harmonic analysis. You can access the article here.

Spektrum der Wissenschaften: A German translation of the Scientific American article (above) appeared in the popular science magazine 'Spektrum der Wissenschaft', and can be found here.

Austrian Broadcasting Agency (ORF): the work of Sam Chow and me on multiplicative Diophantine approximation was featured here (in German). A more complete list of press releases can be found at the website of the ASciNA awards when scrolling to the 2022 awards.

Complete list of published or accepted journal articles:

Ch. Lutsko, and N. Technau: Correlations of Fractional Parts of $\alpha n^\theta$.

to appear in Amer. J. Math.

(Here is the arXiv version.)Ch. Lutsko, and N. Technau: Full Poissonian Local Statistics of Slowly Growing Sequences.

to appear in Compositio Math.

(Here is the arXiv version.)S. Chow, and N. Technau: Dispersion and Littlewood's Conjecture.

Advances Math., DOI: 10.1016/j.aim.2024.109697.Ch. Lutsko, A. Sourmelidis, and N. Technau: Pair Correlations of Fractional Parts of $\alpha n^\theta$.

J. Eur. Math. Soc. DOI: 10.4171/jems/1449.

(Here is the arXiv version.)S. Chow, and N. Technau: Counting multiplicative approximations.

Ramanujan J. 62 (2023), pp. 241–250, DOI: 10.1007/s11139-022-00610-3.

(Here is the arXiv version.)Z. Rudnick, and N. Technau: The metric theory of the pair correlation function for small non-integer powers.

J. London Math. Soc., DOI: 10.1112/jlms.12647.

(Here is the arXiv version.)N. Technau, and N. Yesha: On the correlations of $n^\alpha$ mod 1.

J. Eur. Math. Soc. 25 (2023), no. 10, pp. 4123–4154, DOI: 10.4171/JEMS/1281.

(Here is the arXiv version.)F. Pazuki, N. Technau, and M. Widmer: Northcott numbers for the house and the Weil height.

Bull. London Math. Soc., DOI: 10.1112/blms.12662.

(Here is the arXiv version.)Ch. Aistleitner, S. Baker, N. Technau, and N. Yesha: Gap statistics and higher correlations for geometric progressions modulo one.

Math. Ann. 385 (2023), pp. 845–861, DOI: 10.1007/s00208-022-02362-3.

(Here is the arXiv version.)Ch. Aistleitner, N. Technau, and A. Zafeiropoulos: On the order of magnitude of Sudler products.

Amer. J. Math. 145 (2023), pp. 721-764 DOI: 10.1353/ajm.2023.a897495.

(Here is the arXiv version.)N. Technau, and A. Walker: On the triple correlations of fractional parts of $n^2 \alpha$,

Canadian J. Math., DOI: 10.4153/s0008414x21000249,

(Here is the arXiv version.)Z. Rudnick, and N. Technau: The metric theory of the pair correlation function of real-valued lacunary sequences,

Illinois J. Math., 64, pp. 583–594 (2020).

(Here is the journal version and arXiv version.)N. Technau, and A. Zafeiropoulos: The discrepancy of $(n_k x)_{k=1}^{\infty}$ with respect to certain probability measures,

Q. J. Math., 71, pp. 573–597 (2020).

(Here is the journal version and arXiv version.)N. Technau, and M. Widmer: Counting lattice points and weak admissibility of a lattice and its dual,

Israel J. Math., pp. 1–19 (2020).

(Here is the journal version and arXiv version.)Ch. Aistleitner, T. Lachmann, M. Munsch, N. Technau, and A. Zafeiropoulos: The Duffin-Schaeffer conjecture with extra divergence,

Advances Math., 356 (2019), Article ID 106808.

(Here is the journal version and arXiv version.)S. Chow, and N. Technau: Higher-rank Bohr sets and multiplicative diophantine approximation,

Compos. Math., 155, pp. 2214–2233 (2019).

(Here is the journal version and arXiv version.)C. Elsner, and N. Technau: On linear relations for Dirichlet series formed by recursive sequences of second order,

J. Aust. Math, pp. 1–25 (2020).

(Here is the journal version and arXiv version.)Ch. Aistleitner, T. Lachmann, and N. Technau: There is no Khintchine threshold for metric pair correlations,

Mathematika 65, pp. 990–1009 (2019).

(Here is the journal version and arXiv version.)Ch. Elsholtz, M. Technau, and N. Technau: The maximal order of iterated multiplicative functions,

Mathematika 65, pp. 929–949 (2019).

(Here is the journal version and arXiv version.)T. Lachmann, and N. Technau: On exceptional sets in the metric Poissonian pair correlations problem,

Monatsh. Math., pp. 1–20 (2017).

(Here is the journal version and arXiv version.)Ch. Elsholtz, N. Technau, and R. Tichy: On the regularity of primes in arithmetic progressions,

Int. J. Number Theory 13, pp. 1349–1361 (2017).

(Here is the journal version and arXiv version.)M. Technau, and N. Technau: A Loewner equation for infinitely many slits,

Comput. Methods Funct. Theory 17, pp. 255–272 (2017).

(Here is the journal version.)

Published/Accepted books:

S. Chow, and N. Technau: Littlewood and Duffin--Schaeffer type problems in diophantine approximation,

Mem. Amer. Math. Soc. (2024), DOI: https://doi.org/10.1090/memo/1475.

(Here is the arXiv version.)

Submitted manuscripts and manuscripts under revision:

S. Chow, and N. Technau: Smooth discrepancy and Littlewood's conjecture.

(Here is the arXiv version.)M. Radziwiłł, and N. Technau: The gap distribution of $\sqrt{n} mod 1$ and the circle method.

(Here is the arXiv version.)D. Schindler, R. Srivastava, and N. Technau: Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals.

(Here is the arXiv version.)R. Srivastava, and N. Technau: Density of Rational Points Near Flat/Rough Hypersurfaces.

(Here is the arXiv version.)

Here is a list of questions/problems/conjectures that these papers resolve:

Aistleitner, Larcher, Eddin, Tichy: about the growth of Sudler products, see 2002.06602 on the arXiv.

Aistleitner, Baker: on the gap distribution of geometric progressions modulo one, see 2010.10355 on the arXiv.

Aistleitner, El-Baz, Munsch (unofficially Rudnick asked this question first): about the metric pair correlation of small integer powers, see 2107.07092 on the arXiv.

Beresnevich, Haynes, Velani: on a higher dimensional fibre-version of Gallgher's theorem in multiplicative Diophantine approximation, see 1810.04558 on the arXiv.

Beresnevich, Haynes, Velani: on a fully inhomogeneous higher dimensional fibre-version of Gallgher's theorem, see 2010.09069 on the arXiv.

Bloom, Chow, Gafni, Walker: concerning a Khintchine-like threshold for metric Poissonian pair correlations, see 1802.02659 on the arXiv.

Grepstad, Kaltenböck, Neumüller: concerning the growth of Sudler products, see 2002.06602 on the arXiv.

Haynes, Pollington, Velani: on the Duffin-Schaeffer conjecture with an extra divergence assumption, see 1803.05703 on the arXiv.

Haynes, Jensen, Kristensen: about the magnitude of the discrepancy of lacunary sequences dilated by badly approximable numbers, see 1812.06293 on the arXiv.

M. Stein: about the linear independence of even values of the certain Dirichlet series, see 1805.03003 on the arXiv.

Steinerberger: on the relation between low discrepancy sequences and the number variance, see 2005.01490 on the arXiv.

Here is a list of obstructions or thresholds that these papers break through:

The Rudnick-Sarnak obstruction for the triple correlations of dilated squares, see 2005.01490 on the arXiv.

Here is a list of substantial progress to resolving questions/empirical observations that were made:

Marklof (ICERM talk, p. 54): empirical observation on the Poissonian level spacing of (generalised) monomials of slow growth, see 2106.09800 on the arXiv.

Beresnevich: About the number of rational points close to non-analytic and non-degenerate manifolds, see 2310.03867 on the arXiv.