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Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. HareNikita Sidorov — 2015

Studia Mathematica

Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if | d e t M | 2 - 1 / d , then the attractor A M has non-empty interior. We also consider the set M of points in A M which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of M is positive. For this special class the full description of M is given as well. This paper continues our work begun...

Patterns and periodicity in a family of resultants

Kevin G. HareDavid McKinnonChristopher D. Sinclair — 2009

Journal de Théorie des Nombres de Bordeaux

Given a monic degree N polynomial f ( x ) [ x ] and a non-negative integer , we may form a new monic degree N polynomial f ( x ) [ x ] by raising each root of f to the th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

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