26,643 research outputs found
Sparse regular subsets of the reals
This paper concerns the expansion of the real ordered additive group by a
predicate for a subset of whose base- representations are recognized
by a B\"uchi automaton. In the case that this predicate is closed, a dichotomy
is established for when this expansion is interdefinable with the structure
(\mathbb{R},1}. In the
case that the closure of the predicate has Hausdorff dimension less than ,
the dichotomy further characterizes these expansions of
by when they have NIP and NTP, which is precisely when the closure of the
predicate has Hausdorff dimension .Comment: 25 page
The representation of integers by positive ternary quadratic polynomials
An integral quadratic polynomial is called regular if it represents every
integer that is represented by the polynomial itself over the reals and over
the -adic integers for every prime . It is called complete if it is of
the form , where is an integral quadratic
form in the variables and is a
vector in . Its conductor is defined to be the smallest positive
integer such that . We prove that for a
fixed positive integer , there are only finitely many equivalence classes of
positive primitive ternary regular complete quadratic polynomials with
conductor . This generalizes the analogous finiteness results for positive
definite regular ternary quadratic forms by Watson and for ternary triangular
forms by Chan and Oh
Distinguishing subgroups of the rationals by their Ramsey properties
A system of linear equations with integer coefficients is partition regular
over a subset S of the reals if, whenever S\{0} is finitely coloured, there is
a solution to the system contained in one colour class. It has been known for
some time that there is an infinite system of linear equations that is
partition regular over R but not over Q, and it was recently shown (answering a
long-standing open question) that one can also distinguish Q from Z in this
way.
Our aim is to show that the transition from Z to Q is not sharp: there is an
infinite chain of subgroups of Q, each of which has a system that is partition
regular over it but not over its predecessors. We actually prove something
stronger: our main result is that if R and S are subrings of Q with R not
contained in S, then there is a system that is partition regular over R but not
over S. This implies, for example, that the chain above may be taken to be
uncountable.Comment: 14 page
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