57 research outputs found
The triangular theorem of eight and representation by quadratic polynomials
We investigate here the representability of integers as sums of triangular
numbers, where the -th triangular number is given by . In
particular, we show that ,
for fixed positive integers , represents every nonnegative
integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if
`cross-terms' are allowed in , we show that no finite set of positive
integers can play an analogous role, in turn showing that there is no
overarching finiteness theorem which generalizes the statement from positive
definite quadratic forms to totally positive quadratic polynomials
Finding simultaneous Diophantine approximations with prescribed quality
We give an algorithm that finds a sequence of approximations with Dirichlet
coefficients bounded by a constant only depending on the dimension. The
algorithm uses the LLL-algorithm for lattice basis reduction. We present a
version of the algorithm that runs in polynomial time of the input.Comment: 16 pages, 7 figure
A remarkable sequence related to and
We prove that five ways to define entry A086377 in the On-Line Encyclopedia
of Integer Sequences do lead to the same integer sequence
Entropy quotients and correct digits in number-theoretic expansions
Expansions that furnish increasingly good approximations to real numbers are
usually related to dynamical systems. Although comparing dynamical systems
seems difficult in general, Lochs was able in 1964 to relate the relative speed
of approximation of decimal and regular continued fraction expansions (almost
everywhere) to the quotient of the entropies of their dynamical systems. He
used detailed knowledge of the continued fraction operator. In 2001, a
generalization of Lochs' result was given by Dajani and Fieldsteel in
\citeDajF, describing the rate at which the digits of one number-theoretic
expansion determine those of another. Their proofs are based on covering
arguments and not on the dynamics of specific maps. In this paper we give a
dynamical proof for certain classes of transformations, and we describe
explicitly the distribution of the number of digits determined when comparing
two expansions in integer bases. Finally, using this generalization of Lochs'
result, we estimate the unknown entropy of certain number theoretic expansions
by comparing the speed of convergence with that of an expansion with known
entropy.Comment: Published at http://dx.doi.org/10.1214/074921706000000202 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
The -Class Tower of
The seminal papers in the field of root-discriminant bounds are those of
Odlyzko and Martinet. Both papers include the question of whether the field
has finite or infinite -class tower. This is a
critical case that will either substantially lower the best known upper bound
for lim inf of root-discriminants (if infinite) or else give a counter-example
to what is often termed Martinet's conjecture or question (if finite). Using
extensive computation and introducing some new techniques, we give strong
evidence that the tower is in fact finite, establishing other properties of its
Galois group en route
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