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The integer Chebyshev constant of Farey intervals.

Julián Aguirre, Juan Carlos Peral (2007)

Publicacions Matemàtiques

We obtain new bounds for the integer Chebyshev constant of intervals [p/q, r/s] where p, q, r and s are non-negative integers such that qr - ps = 1. As a consequence of the methods used, we improve the known lower bound for the trace of totally positive algebraic integers.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

Let M ( α ) be the Mahler measure of an algebraic number α , and V be an open subset of . Then its Lehmer constant L ( V ) is inf M ( α ) 1 / deg ( α ) , the infimum being over all non-zero non-cyclotomic α lying with its conjugates outside V . We evaluate L ( V ) when V is any annulus centered at 0 . We do the same for a variant of L ( V ) , which we call the transfinite Lehmer constant L ( V ) .Also, we prove the converse to Langevin’s Theorem, which states that L ( V ) > 1 if V contains a point of modulus 1 . We prove the corresponding result for L ( V ) .

The R₂ measure for totally positive algebraic integers

V. Flammang (2016)

Colloquium Mathematicae

Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates α = α , . . . , α d are positive real numbers. We study the set ₂ of the quantities ( i = 1 d ( 1 + α ² i ) 1 / 2 ) 1 / d . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A G l d ( ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u 2 with A u : = A | E u G L 2 ( ) as its expansion. In the last section we find conditions on A so...

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