Arithmetik an der A7 in Graz ()
About the “AaA7”
The “Arithmetik an der A7” — “AaA7” for short — is a biannual meeting of the number theory research groups from Hannover, Hildesheim, Würzburg and Ulm with regular attendence of colleagues based at various other institutions. The venue of the conference usually alternates between the aforementioned locations, but in the past the conference has also been hosted in Berlin and Regensburg, for instance. This time, the AaA7 takes place in Austria for the second time, following a previous iteration in Linz in 2020.
There is no conference fee. Anyone interested in attending is most welcome to participate. For organisational reasons we do, however, ask you to register.
The present iteration of the conference is hosted at the Institute of Analysis and Number Theory of Graz University of Technology.
The local organisers are Nick Rome, Athanasios Sourmelidis, and Marc Technau.
Funding
The conference is financially supported by the FWF (“Fonds zur Förderung der wissenschaftlichen Forschung”, projects I 4945-N, M 3246-N and ESP 441-NBL) and the ANR (“Agence nationale de la recherche”, project ANR-20-CE91-0006).
While each participant is kindly requested to seek reimbursement of their expenses from their respective home institution as a first option, limited funding for reimbursing parts of the incurred travelling expenses of the speakers is available. If you are interested in giving a talk, and would like us to reimburse parts of your expenses, please inform us accordingly in your registration email. In this case, if possible, please also give us a rough estimate of your expenses and be prepared to tell us the exact amount at the conference.
Speakers
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Christian Bernert (Leibniz University Hannover)
Cubic forms over imaginary quadratic number fields (and why we should care)Abstract: I will report on joint work with Leonhard Hochfilzer, showing that cubic forms in at least 14 variables over imaginary quadratic number fields always admit a non-trivial solution, generalizing a result of Heath-Brown over the rationals. The proof uses a version of the circle method. As pioneered by Wooley, this result has applications to cubic forms over the rationals, where we conclude that any system of two cubic forms in at least 627 variables has a rational point and (together with recent work of Brandes–Dietmann) that any cubic hypersurface in at least 31 variables contains a rational line, in both cases improving on previous results.
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Rainer Dietmann (Royal Holloway, University of London)
Sieving with square conditions and applications to Hilbert cubes in arithmetic setsAbstract: I want to report on recent joint work with Christian Elsholtz and Imre Ruzsa on applications of Gallagher's larger sieve with initial sieving conditions modulo squares of primes which via a new combinatorial lemma can be transformed to more suitable conditions modulo primes. As an application we obtain new upper bounds for the dimension of Hilbert cubes in certain arithmetic sets such as the set of powerful numbers.
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Ramūnas Garunkštis (Vilnius University)
Discrete mean value theorems for the Riemann zeta-function over its shifted nontrivial zerosAbstract: Let \(\zeta(s)\) be the Riemann zeta function and \(\rho=\beta+i\gamma\) its nontrivial zero. In 1984 S. M. Gonek started to consider the mean value \[\sum_{0<\gamma<T}\zeta(\rho+\delta)\mkern 1mu\zeta(1-\rho+\overline{\delta})\] for imaginary \(\delta\). Here we present asymptotic formulas for this mean value for general complex \(\delta\).
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Pascal Jelinek (University of Leoben)
Square-free values of polynomials on averageAbstract: The number of square-free integers in \(N\) consecutive values of any polynomial \(f\) is conjectured to be \(c_fN\), where the constant \(c_f\) depends only on the polynomial \(f\). This has been proven for degree less or equal to 3. In 2013, Igor Shparlinski proved that this conjecture holds on average over all polynomials of a fixed naive height. In my talk I will describe a new general method first developed by Timothy Browning, Efthymios Sofos and Joni Teräväinen, and derive new results with less averaging.
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Sumaia Saad Eddin (Austrian Academy of Sciences, RICAM)
Asymptotic results for a class of arithmetic functions involving the greatest common divisorAbstract: In analytic number theory, arithmetic functions are analyzed using techniques from analysis and complex analysis. Properties of these functions, such as their growth rates and asymptotic behavior, provide insights into the behavior of prime numbers and other integer sequences. In this talk, we will present recent results on the asymptotic behavior of a class of arithmetic functions that involve the greatest common divisor.
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Lukas Spiegelhofer (University of Leoben)
Thue–Morse along the sequence of cubesAbstract: Let \(t=01101001\!\cdots\) be the Thue–Morse word. This automatic sequence can be defined as the binary sum-of-digits function along the set of natural numbers, reduced modulo 2. We prove that the asymptotic density of the set of natural numbers $n$ satisfying \(t(n^3)=0\) equals \(1/2\). Comparable results, featuring asymptotic equivalence along a polynomial as in our theorem, were previously only known for the linear case, and for the sequence of squares [Mauduit–Rivat 2009].
Concerning the Thue–Morse sequence along polynomials \(p\) of degree at least three, previous results were restricted to inequalities, such as a lower bound for the number of integers \(n\) satisfying \(t(p(n))=0\) (while there exist sharper theorems concerning the sum-of-digits function with respect to “large bases”). By proving an asymptotic equivalence for the case of a cubic polynomial, we move one step closer to the solution of the third Gelfond problem on the sum-of-digits function [Gelfond 1967/1968]. -
Helmut Maier (Ulm University)
The problem of propinquity of divisors for Gaussian integersAbstract: The problem of the propinquity of divisors of an integer was treated in 1984 in joint work of the speaker with G. Tenenbaum. The speaker wants to report on joint work with Saurabh Kumar Singh on the analogous problem for Gaussian integers.
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Roland Miyamoto
Fixed points of stribolic operators and their combinatorial aspectsAbstract: Levine's popular sequence (A011784) points us to a real function in the first quadrant which becomes its own derivative when turned \(90^\circ\) clockwise (into the fourth quadrant) and can be viewed as fixed point of a “stribolic” operator. While traditional fixed-point theorems seem inadequate, we employ geometric techniques to prove constructive and global convergence for the rescaling stribolic operator \(T\) to its fixed function \(h\) via its canonical iterates \(h_n := T^n {\boldsymbol{1}}_{[0,1]}\). Our algorithmic approximation of \(h\) and its associated “stribolic” constant \(\kappa\approx0.27887706137\) involves polynomials over \(\mathbb{Q}\) of Fibonacci degree. Time permitting, higher order stribolic operators and their fixed points are investigated.
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Robert F. Tichy (Graz University of Technology)
Pseudorandom measuresAbstract: In 1997 C. Mauduit and A. Sarkőzy introduced various measures for pseudo randomness of binary sequences over \(\{-1, +1\}\). In this talk the focus lies on the correlation measure of order \(s\). If \(E_{N}=\{e_{1},\ldots, e_{N}\}\) is a finite binary sequence, then the correlation measure of order \(s\) of \(E_{N}\) is defined as \[ C_{s}(E_{N}) = \max_{M,\mkern 2mu\underline{d}} \biggl\lvert \sum^{M}_{n=1}e_{n+d_{1}}e_{n+d_{2}}\ldots e_{n+d_{s}} \biggr\rvert, \] where the maximum is taken over all positive integers \(M\leq N\) and all correlation vectors \(\underline{d}= (d_{1},\ldots,d_{s})\in \mathbb{N}^{s}\) with \(0 \leq d_{1} < d_{2} < \ldots < d_{s} \leq N - M\). We construct binary sequences from certain functions \(f(x)\) of polynomial growth by \(e_{n}= \chi (f(n)),\) where \[ \chi(x) = \begin{cases} +1 & \text{for } 0 \leq \{x\} < \frac{1}{2} \\ -1 & \text{for } \frac{1}{2}\leq \{x\} < 1. \end{cases} \] The approach works for functions \(f(x)= x^{c}\) (\(c > 1\) not an integer) and for Hardy fields as introduced by Boshernitzan (1994) to equidistribution theory. The main result is an estimate of the form \(C_{s}(E_{N})\ll N^{1-\eta}\) for some constant \(\eta>0\). In contrary, we have \(C_{2}(E_{N})\gg N\) for \(f(x) = x^{c}\) with \(0< c < 1\), whereas such sequences are still uniformly distributed modulo 1.
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Christian Weiß (Ruhr West University of Applied Sciences)
Uniform distribution — the \(p\)-adic viewpointAbstract: Real uniformly distributed sequences have been widely studied for many years now. However, the notion of uniform distribution does not only make sense for real numbers but also in a \(p\)-adic setting. Also other definitions like discrepancy and the pair correlation statistic can be transferred to this case. There are interesting parallels and differences to the real case which will be the main focus of the talk.
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Martin Widmer (Royal Holloway, University of London)
A quantitative version of the primitive element theorem for number fieldsAbstract: Each number field is a simple extension of \(\mathbf{Q}\), and thus generated by a single element \(\alpha\). How big is the smallest such \(\alpha\)? In 1998 Wolfgang Ruppert proposed the following precise version of this problem. Given an integer \(d>1\), does there exists a constant \(c_d>0\) such that each number field \(K\) of degree \(d\) has a primitive element whose minimal polynomial over \(\mathbf{Z}\) has Mahler measure at most \(c_d\sqrt{\lvert\Delta_K\rvert}\)? Here \(\Delta_K\) denotes the discriminant of the field \(K\). Ruppert's question is still open but we report on progress that has been made over the years. This is joint work with Shabnam Akhtari and Jeff Vaaler.
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Shuntaro Yamagishi (Institute of Science and Technology Austria)
Rational curves on complete intersectionsAbstract: In this talk, I will present our work (joint with T. Browning and P. Vishe) on the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. This is done by using the circle method over function fields.
Programme
Thursday
- 14:00–14:10:
- Opening
- 14:10–14:40:
- Lukas Spiegelhofer:
Thue–Morse along the sequence of cubes - 14:40–15:10:
- Pascal Jelinek:
Square-free values of polynomials on average - 15:10–15:40:
- Helmut Maier:
The problem of propinquity of divisors for Gaussian integers - 15:40–16:00:
- Coffee and tea break
- 16:00–16:30:
- Martin Widmer:
A quantitative version of the primitive element theorem for number fields - 16:30–17:00:
- Sumaia Saad Eddin:
Asymptotic results for a class of arithmetic functions involving the greatest common divisor - 17:00–17:30:
- Christian Weiß:
Uniform distribution — the \(p\)-adic viewpoint - 18:00–:
- Dinner at “Restaurant Brandhof” (Gleisdorfergasse 10, 8010 Graz)
Friday
- 09:00–09:20:
- Coffee and tea break
- 09:20–09:50:
- Ramūnas Garunkštis:
Discrete mean value theorems for the Riemann zeta-function over its shifted nontrivial zeros - 09:50–10:20:
- Rainer Dietmann:
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets - 10:20–10:50:
- Roland Miyamoto:
Fixed-points of stribolic operators and their combinatorial aspects - 10:50–11:20:
- Robert F. Tichy:
Pseudorandom measures - 11:20–11:40:
- Coffee and tea break
- 11:40–12:10:
- Christian Bernert:
Cubic forms over imaginary quadratic number fields (and why we should care) - 12:10–12:40:
- Shuntaro Yamagishi:
Rational curves on complete intersections - 12:45–12:50:
- Group photo
- 12:50–:
- Tour through Graz
Participants
- Christoph Aistleitner
- Christian Bernert
- Dmitriy Bilyk
- Rainer Dietmann
- Jasmin Fiedler
- Lorenz Frühwirth
- Jakob Führer
- Ramūnas Garunkštis
- Valentin Havlovec
- Sebastian Heintze
- Pascal Jelinek
- Helmut Maier
- Roland Miyamoto
- Christoph Raunjak
- Nick Rome
- Sumaia Saad Eddin
- Andrei Shubin
- Athanasios Sourmelidis
- Lukas Spiegelhofer
- Eduard Stefanescu
- Marc Technau
- Robert Tichy
- Matteo Verzobio
- Christian Weiß
- Martin Widmer
- Lena Wurzinger
- Shuntaro Yamagishi