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Jan-Christoph Schlage-Puchta

Publications and source records attributed to Jan-Christoph Schlage-Puchta.

At least 37 records · Page 2Linked to original sources
arXiv2018

Totally non congruence Veech groups

Veech groups are discrete subgroups of SL(2, R) which play an important role in the theory of translation surfaces. For a special class of translation surfaces called origamis or square-tiled surfaces their Veech groups are subgroups of finite index of SL(2, Z). We show that each stratum of the space of translation surfaces contains infinitely many origamis whose Veech group is a totally non congruence group, i.e. it surjects to SL(2, Z/nZ) for any n.

math.GT↗
arXiv2017

Davenport's constant for groups with large exponent

Let $G$ be a finite abelian group. We show that its Davenport constant $D(G)$ satisfies $D(G)\leq \exp(G)+\frac{|G|}{\exp(G)}-1$, provided that $\exp(G)\geq\sqrt{|G|}$, and $D(G)\leq 2\sqrt{|G|}-1$, if $\exp(G)<\sqrt{|G|}$. This proves a conjecture by Balasubramanian and the first named author.

math.NT↗
arXiv2013

Finite translation surfaces with maximal number of translations

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.

math.GT↗
arXiv2012

Ramsey numbers R(K3,G) for graphs of order 10

In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.

math.CO↗
arXiv2012

On $p$-deficiency in groups

Recently, Schlage-Puchta proved super multiplicity of $p$-deficiency for normal subgroups of $p$-power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive $p$-deficiency have virtually positive $p$-deficiency. We also compute the $p$-deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute $p$-homology gradients and the $p$-Euler characteristic.

math.GR↗
arXiv2011

On Fabry's Gap Theorem

By combining Turán's proof of Fabry's gap theorem with a gap theorem of P. Szüsz we obtain a gap theorem which is more general then both these theorems.

math.CV↗
arXiv2011

A $p$-group with positive Rank Gradient

We construct for $d\geq 2$ and $ε>0$ a $d$-generated $p$-group $Γ$, which in an asymptotic sense behaves almost like a $d$-generated free pro-$p$-group. We show that a subgroup of index $p^n$ needs $(d-ε)p^n$ generators, and that the subgroup growth of $Γ$ satisfies $s_{p^n}(Γ)>s_{p^n}(F_d^p)^{1-ε}$, where $F_d^p$ is the $d$-generated free pro-$p$-group. To do so we introduce a new invariant for finitely generated groups and study some of its basic properties.

math.GR↗
arXiv2011

On triangular billiards

We prove a conjecture of Kenyon and Smillie concerning the nonexistence of acute rational-angled triangles with the lattice property.

math.DS↗
arXiv2011

An additive property of almost periodic sets

We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method.

math.NT↗
arXiv2011

Sign changes of $π(x, q, 1) - π(x, q, a)$

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $π(x, q, 1) - π(x, q, a)$ has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming the Riemannian hypothesis we give a lower bound for the number of sign changes of $π(x)-\li x$. The implied results for the least sign change are weaker then those obtained by numerical methods, however, our method makes no use of computations of zeros of the $ζ$-function.

math.NT↗
arXiv2011

The order of elements in Sylow $p$-subgroups of the symmetric group

Define a random variable $ξ_n$ by choosing a conjugacy class $C$ of the Sylow $p$-subgroup of $S_{p^n}$ by random, and let $ξ_n$ be the logarithm of the order of an element in $C$. We show that $ξ_n$ has bounded variance and mean order $\frac{\log n}{\log p}+O(1)$, which differs significantly from the average order of elements chosen with equal probability.

math.GR↗
arXiv2011

The equation $ω(n)=ω(n+1)$

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.

math.NT↗
arXiv2011

Primes in short arithmetic progressions

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg sieve.

math.NT↗
arXiv2011

On Shanks' Algorithm for Modular Square Roots

Let $p$ be a prime number, $p=2^nq+1$, where $q$ is odd. D. Shanks described an algorithm to compute square roots $\pmod{p}$ which needs $O(\log q + n^2)$ modular multiplications. In this note we describe two modifications of this algorithm. The first needs only $O(\log q + n^{3/2})$ modular multiplications, while the second is a parallel algorithm which needs $n$ processors and takes $O(\log q+n)$ time.

math.NT↗