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Chris Pinner

Publications and source records attributed to Chris Pinner.

16 recordsLinked to original sources
arXiv2025

The integer group determinants for $GA(1,q)$

We show that the integer group determinants for the general affine group of degree one, $GA(1,q)$ with $q=p^k$ a prime power, take the form $D=AB^{q-1},$ where $A$ is a $\mathbb Z_{q-1}$ integer group determinant and $B\equiv A \bmod q$. This generalizes the result for $k=1$. When $2^k-1$ is a Mersenne prime we show that this condition is both necessary and sufficient for $GA(1,2^k).$ The same is true for $GA(1,9)$ and $GA(1,27)$, but interestingly not for $GA(1,16)$.

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arXiv2024

The integer group determinants for the abelian groups of order 18

We obtain a complete description of the integer group determinants for $\mathbb Z_{18}$ (these are the $18\times18$ circulant determinants with integer entries) and $\mathbb Z_3 \times \mathbb Z_6$, the two abelian groups of order 18. This completes the groups of order less than 20.

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arXiv2024

The integer group determinants for GA(1,p) and related semidirect products

We consider the integer group determinants for groups that are semidirect products of $\mathbb Z_p$ and $\mathbb Z_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine groups of degree one GA(1,$p$) when $p=5,7,11$ and $23$, and for $\mathbb Z_7\rtimes \mathbb Z_3,$ $\mathbb Z_{11}\rtimes \mathbb Z_5$ and $\mathbb Z_{13}\rtimes \mathbb Z_6,$ showing that the obvious divisibility and congruence conditions arising from the form of the group determinant when $n=p-1$ or $\frac{1}{2}(p-1)$, can be sufficient as well as necessary for these types of groups (although in the latter case we must work with norms of integers in a quadratic field). For $p=13$ this also happens for the remaining groups of this type, $\mathbb Z_{13}\rtimes_5 \mathbb Z_4$ and $\mathbb Z_{13}\rtimes \mathbb Z_3$, (working in an appropriate cubic and quartic field).

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arXiv2023

The integer group determinants for the semidihedral group of order 16

We obtain a complete description of the integer group determinants for SmallGroup(16,8), the semidihedral group of order 16. While this paper was in preparation, a complete descriptions for this group was independently obtained by Yuka Yamaguchi and Naoya Yamaguchi in [21] (and the other remaining group of order 16 in [20]). We offer our version here for anyone interested in a slightly different approach.

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arXiv2023

The integer group determinants for SmallGroup(16,13)

We obtain a complete description of the integer group determinants for SmallGroup(16,13), the central product of the dihedral group of order eight and cyclic group of order four. These values are the same as the integer group determinants for SmallGroup(16,11), the direct product of the dihedral group of order eight and cyclic group of order two. It was not previously known that the integer group determinants do not determine the group.

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arXiv2023

An upper bound on the inhomogeneous approximation constants

For an irrational real $\alpha$ and $\gamma\not \in \mathbb Z + \mathbb Z\alpha$ it is well known that $$ \liminf_{|n|\rightarrow \infty} |n| ||n\alpha -\gamma || \leq \frac{1}{4}. $$ If the partial quotients, $a_i,$ in the negative `round-up' continued fraction expansion of $\alpha$ have $R:=\liminf_{i\rightarrow \infty}a_i$ odd, then the 1/4 can be replaced by $$ \frac{1}{4}\left(1-\frac{1}{R}\right)\left(1-\frac{1}{R^2}\right), $$ which is optimal. The optimal bound for even $R\geq 4$ was already known.

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arXiv2023

Bounding the Largest Inhomogeneous Approximation Constant

For a given irrational number $\alpha$ and a real number $\gamma$ in $(0,1)$ one defines the two-sided inhomogeneous approximation constant \begin{equation*} M(\alpha,\gamma):=\liminf_{|n|\rightarrow\infty}|n| ||n\alpha-\gamma||, \end{equation*} and the case of worst inhomogeneous approximation for $\alpha$ \begin{equation*} \rho(\alpha):=\sup_{\gamma\notin\mathbb{Z}+\alpha\mathbb{Z}}M(\alpha,\gamma). \end{equation*} We are interested in lower bounds on $\rho(\alpha)$ in terms of $R:=\liminf_{i\rightarrow\infty}a_i,$ where the $a_i$ are the partial quotients in the negative (i.e.\ the `round-up') continued fraction expansion of $\alpha$. We obtain bounds for any $R\geq 3$ which are best possible when $R$ is even (and asymptotically precise when $R$ is odd). In particular when $R\geq 3$ $$ \rho(\alpha)\geq \cfrac{1}{6\sqrt{3}+8}=\cfrac{1}{18.3923\dots}, $$ and when $R\geq 4$, optimally, $$ \rho(\alpha) \geq \cfrac{1}{4\sqrt{3}+2}=\cfrac{1}{8.9282\ldots}. $$

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arXiv2021

Integer circulant determinants of order 15

We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for $\mathbb Z_{pq}$, the cyclic group of order $pq$. We show that $p^2$ and $q^2$ are not determinants (more generally we show that the classic necessary divisibility conditions are never sufficient when $n$ contains at least two odd primes). We obtain a complete description of the integer group determinants for $\mathbb Z_{15}$ (the smallest unresolved group) and partial results for general $n=3p.$

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arXiv2021

Minimal Group Determinants For Dicyclic Groups

We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.

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arXiv2018

A generalization of the Goresky-Klapper conjecture, Part II

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements from every residue classe mod $n$, once $p$ is sufficiently large. If $f(x)=Ax$ mod $p$, then for each $p$ and $n$ there will be exactly $(1+o(1))\frac{6}{\pi^2}n^2$ readily describable values of $A$ for which the image of some residue class mod $n$ misses at least one residue class mod $n,$ even when $p$ is large relative to $n$. A similar situation holds for $f(x)=Ax^{(p+1)/2}$ mod $p$.

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arXiv2018

A generalization of the Goresky-Klapper conjecture, Part I

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps $f(x)=\pm x$ mod $p$ when $n$ is even and $f(x)=\pm x$ or $\pm x^{(p+1)/2}$ mod $p$ when $n$ is odd.

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