Erratum to: On the Geometry of Cyclic Lattices

Discrete & Computational Geometry, Apr 2015

Lenny Fukshansky, Xun Sun

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Erratum to: On the Geometry of Cyclic Lattices

Discrete Comput Geom Erratum to: On the Geometry of Cyclic Lattices Lenny Fukshansky 0 1 Xun Sun 0 1 0 School of Mathematical Sciences, Claremont Graduate University , Claremont, CA 91711 , USA 1 Department of Mathematics, Claremont McKenna College , 850 Columbia Avenue, Claremont, CA 91711 , USA A mistake in the results of our previous paper is corrected. Theorems 1.1 and 1.2 of our paper [1] are incorrect as stated. Specifically, these theorems assert that the well-rounded (WR) lattices comprise a positive proportion of all cyclic lattices in every dimension. This assertion is based on comparative asymptotic counting estimates for WR cyclic lattices versus all cyclic lattices with successive minima bounded by R as R . While the lower bound on the number of WR cyclic lattices produced in Sect. 4 is correct, the upper bound on the number of all cyclic lattices produced in Sect. 3 (specifically, Lemmas 3.2 and 3.3) is not accurate. Similarly, in the case N = 2, there is a mistake in the estimate on the function h2( R) in the proof of Lemma 5.2, while the rest of Lemma 5.2 is correct. For the purposes of our current results, this means that WR cyclic lattices do not comprise a positive proportion of all cyclic lattices in a given dimension, and hence our results should instead be stated as counting estimates on the number of WR cyclic lattices. This Lenny Fukshansky - implies the following corrected statements of our theorems, which follow directly from the results of Sects. 4 and 5 of [1]. We write WR for lattices which are generated by their minimal vectors, a strictly stronger condition than WR in general. Theorem 1.1, corrected. Let R R>0, then there exists a constant N > 0 depending only on dimension N such that as R . Proof This follows immediately from Lemma 4.3 of [1]. Theorem 1.2, corrected. Let R R>0, then 0.200650... R2 3.742382... R # C2 : 2( ) R, is WR 0.267638... R2 + 1.673031... R. Proof This follows immediately from Lemma 5.2 of [1]. Corollary 1.3, corrected. Let N 2, let SN be an N -cycle, and let CN ( ) be the set of all -invariant full-rank sublattices of ZN . Then The above correction means that SVP and SIVP are not equivalent on a positive proportion of cyclic lattices, however they are equivalent on a positive proportion of WR cyclic lattices spanned by rotations of a single vector. In other words, in the denominator of Eq. (18) in Remark 4.4, CN should be replaced by RN . Finally, in light of this correction, it makes sense to restate Question 2 in Sect. 6 of our paper as: How many WR lattices are there among all -invariant sublattices of ZN for an arbitrary permutation SN ?


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Lenny Fukshansky, Xun Sun. Erratum to: On the Geometry of Cyclic Lattices, Discrete & Computational Geometry, 2015, 971-972, DOI: 10.1007/s00454-015-9698-6