scholarly journals Pseudo Frobenius numbers

2019 ◽  
Vol 37 (2) ◽  
pp. 200-206
Author(s):  
Benjamin Sambale
Keyword(s):  
Frobenius Numbers

Equidistribution of Farey sequences on horospheres in covers of and applications

2019 ◽  
Vol 41 (2) ◽  
pp. 471-493
Author(s):  
BYRON HEERSINK
Keyword(s):  
Diophantine Approximation ◽  
Lattice Points ◽  
Index Subgroup ◽  
Farey Sequence ◽  
Frobenius Numbers ◽  
Farey Sequences ◽  
Approximation Problems ◽  
Primitive Lattice Points ◽  
Finite Index Subgroup ◽  
Statistical Results

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

Limit behaviour of large Frobenius numbers

2007 ◽  
Vol 62 (4) ◽  
pp. 713-725 ◽  
Author(s):  
Jean Bourgain ◽  
Yakov G Sinai
Keyword(s):  
Frobenius Numbers ◽  
Limit Behaviour

scholarly journals Integer Knapsacks: Average Behavior of the Frobenius Numbers

2009 ◽  
Vol 34 (3) ◽  
pp. 698-705 ◽  
Author(s):  
Iskander Aliev ◽  
Martin Henk
Keyword(s):  
Average Behavior ◽  
Frobenius Numbers

Euler–Frobenius numbers

2013 ◽  
Vol 24 (10) ◽  
pp. 817-830 ◽  
Author(s):  
Wolfgang Gawronski ◽  
Thorsten Neuschel
Keyword(s):  
Frobenius Numbers
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