In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem:
Problem 1 (Erdös-Ulam problem) Let
be a set such that the distance between any two points in
is rational. Is it true that
cannot be (topologically) dense in
?
The paper of Anning and Erdös addressed the case that all the distances between two points in were integer rather than rational in the affirmative.
The Erdös-Ulam problem remains open; it was discussed recently over at Gödel’s lost letter. It is in fact likely (as we shall see below) that the set in the above problem is not only forbidden to be topologically dense, but also cannot be Zariski dense either. If so, then the structure of
is quite restricted; it was shown by Solymosi and de Zeeuw that if
fails to be Zariski dense, then all but finitely many of the points of
must lie on a single line, or a single circle. (Conversely, it is easy to construct examples of dense subsets of a line or circle in which all distances are rational, though in the latter case the square of the radius of the circle must also be rational.)
The main tool of the Solymosi-de Zeeuw analysis was Faltings’ celebrated theorem that every algebraic curve of genus at least two contains only finitely many rational points. The purpose of this post is to observe that an affirmative answer to the full Erdös-Ulam problem similarly follows from the conjectured analogue of Falting’s theorem for surfaces, namely the following conjecture of Bombieri and Lang:
Conjecture 2 (Bombieri-Lang conjecture) Let
be a smooth projective irreducible algebraic surface defined over the rationals
which is of general type. Then the set
of rational points of
is not Zariski dense in
.
In fact, the Bombieri-Lang conjecture has been made for varieties of arbitrary dimension, and for more general number fields than the rationals, but the above special case of the conjecture is the only one needed for this application. We will review what “general type” means (for smooth projective complex varieties, at least) below the fold.
The Bombieri-Lang conjecture is considered to be extremely difficult, in particular being substantially harder than Faltings’ theorem, which is itself a highly non-trivial result. So this implication should not be viewed as a practical route to resolving the Erdös-Ulam problem unconditionally; rather, it is a demonstration of the power of the Bombieri-Lang conjecture. Still, it was an instructive algebraic geometry exercise for me to carry out the details of this implication, which quickly boils down to verifying that a certain quite explicit algebraic surface is of general type (Theorem 4 below). As I am not an expert in the subject, my computations here will be rather tedious and pedestrian; it is likely that they could be made much slicker by exploiting more of the machinery of modern algebraic geometry, and I would welcome any such streamlining by actual experts in this area. (For similar reasons, there may be more typos and errors than usual in this post; corrections are welcome as always.) My calculations here are based on a similar calculation of van Luijk, who used analogous arguments to show (assuming Bombieri-Lang) that the set of perfect cuboids is not Zariski-dense in its projective parameter space.
We also remark that in a recent paper of Makhul and Shaffaf, the Bombieri-Lang conjecture (or more precisely, a weaker consequence of that conjecture) was used to show that if is a subset of
with rational distances which intersects any line in only finitely many points, then there is a uniform bound on the cardinality of the intersection of
with any line. I have also recently learned (private communication) that an unpublished work of Shaffaf has obtained a result similar to the one in this post, namely that the Erdös-Ulam conjecture follows from the Bombieri-Lang conjecture, plus an additional conjecture about the rational curves in a specific surface.
Let us now give the elementary reductions to the claim that a certain variety is of general type. For sake of contradiction, let be a dense set such that the distance between any two points is rational. Then
certainly contains two points that are a rational distance apart. By applying a translation, rotation, and a (rational) dilation, we may assume that these two points are
and
. As
is dense, there is a third point of
not on the
axis, which after a reflection we can place in the upper half-plane; we will write it as
with
.
Given any two points in
, the quantities
are rational, and so by the cosine rule the dot product
is rational as well. Since
, this implies that the
-component of every point
in
is rational; this in turn implies that the product of the
-coordinates of any two points
in
is rational as well (since this differs from
by a rational number). In particular,
and
are rational, and all of the points in
now lie in the lattice
. (This fact appears to have first been observed in the 1988 habilitationschrift of Kemnitz.)
Now take four points ,
in
in general position (so that the octuplet
avoids any pre-specified hypersurface in
); this can be done if
is dense. (If one wished, one could re-use the three previous points
to be three of these four points, although this ultimately makes little difference to the analysis.) If
is any point in
, then the distances
from
to
are rationals that obey the equations
for , and thus determine a rational point in the affine complex variety
defined as
By inspecting the projection from
to
, we see that
is a branched cover of
, with the generic cover having
points (coming from the different ways to form the square roots
); in particular,
is a complex affine algebraic surface, defined over the rationals. By inspecting the monodromy around the four singular base points
(which switch the sign of one of the roots
, while keeping the other three roots unchanged), we see that the variety
is connected away from its singular set, and thus irreducible. As
is topologically dense in
, it is Zariski-dense in
, and so
generates a Zariski-dense set of rational points in
. To solve the Erdös-Ulam problem, it thus suffices to show that
Claim 3 For any non-zero rational
and for rationals
in general position, the rational points of the affine surface
is not Zariski dense in
.
This is already very close to a claim that can be directly resolved by the Bombieri-Lang conjecture, but is affine rather than projective, and also contains some singularities. The first issue is easy to deal with, by working with the projectivisation
of , where
is the homogeneous quadratic polynomial
with
and the projective complex space is the space of all equivalence classes
of tuples
up to projective equivalence
. By identifying the affine point
with the projective point
, we see that
consists of the affine variety
together with the set
, which is the union of eight curves, each of which lies in the closure of
. Thus
is the projective closure of
, and is thus a complex irreducible projective surface, defined over the rationals. As
is cut out by four quadric equations in
and has degree sixteen (as can be seen for instance by inspecting the intersection of
with a generic perturbation of a fibre over the generically defined projection
), it is also a complete intersection. To show (3), it then suffices to show that the rational points in
are not Zariski dense in
.
Heuristically, the reason why we expect few rational points in is as follows. First observe from the projective nature of (1) that every rational point is equivalent to an integer point. But for a septuple
of integers of size
, the quantity
is an integer point of
of size
, and so should only vanish about
of the time. Hence the number of integer points
of height comparable to
should be about
this is a convergent sum if ranges over (say) powers of two, and so from standard probabilistic heuristics (see this previous post) we in fact expect only finitely many solutions, in the absence of any special algebraic structure (e.g. the structure of an abelian variety, or a birational reduction to a simpler variety) that could produce an unusually large number of solutions.
The Bombieri-Lang conjecture, Conjecture 2, can be viewed as a formalisation of the above heuristics (roughly speaking, it is one of the most optimistic natural conjectures one could make that is compatible with these heuristics while also being invariant under birational equivalence).
Unfortunately, contains some singular points. Being a complete intersection, this occurs when the Jacobian matrix of the map
has less than full rank, or equivalently that the gradient vectors
for are linearly dependent, where the
is in the coordinate position associated to
. One way in which this can occur is if one of the gradient vectors
vanish identically. This occurs at precisely
points, when
is equal to
for some
, and one has
for all
(so in particular
). Let us refer to these as the obvious singularities; they arise from the geometrically evident fact that the distance function
is singular at
.
The other way in which could occur is if a non-trivial linear combination of at least two of the gradient vectors vanishes. From (2), this can only occur if for some distinct
, which from (1) implies that
for two choices of sign . If the signs are equal, then (as
are in general position) this implies that
, and then we have the singular point
If the non-trivial linear combination involved three or more gradient vectors, then by the pigeonhole principle at least two of the signs involved must be equal, and so the only singular points are (5). So the only remaining possibility is when we have two gradient vectors that are parallel but non-zero, with the signs in (3), (4) opposing. But then (as
are in general position) the vectors
are non-zero and non-parallel to each other, a contradiction. Thus, outside of the
obvious singular points mentioned earlier, the only other singular points are the two points (5).
We will shortly show that the obvious singularities are ordinary double points; the surface
near any of these points is analytically equivalent to an ordinary cone
near the origin, which is a cone over a smooth conic curve
. The two non-obvious singularities (5) are slightly more complicated than ordinary double points, they are elliptic singularities, which approximately resemble a cone over an elliptic curve. (As far as I can tell, this resemblance is exact in the category of real smooth manifolds, but not in the category of algebraic varieties.) If one blows up each of the point singularities of
separately, no further singularities are created, and one obtains a smooth projective surface
(using the Segre embedding as necessary to embed
back into projective space, rather than in a product of projective spaces). Away from the singularities, the rational points of
lift up to rational points of
. Assuming the Bombieri-Lang conjecture, we thus are able to answer the Erdös-Ulam problem in the affirmative once we establish
This will be done below the fold, by the pedestrian device of explicitly constructing global differential forms on ; I will also be working from a complex analysis viewpoint rather than an algebraic geometry viewpoint as I am more comfortable with the former approach. (As mentioned above, though, there may well be a quicker way to establish this result by using more sophisticated machinery.)
I thank Mark Green and David Gieseker for helpful conversations (and a crash course in varieties of general type!).
Remark 5 The above argument shows in fact (assuming Bombieri-Lang) that sets
with all distances rational cannot be Zariski-dense, and thus (by Solymosi-de Zeeuw) must lie on a single line or circle with only finitely many exceptions. Assuming a stronger version of Bombieri-Lang involving a general number field
, we obtain a similar conclusion with “rational” replaced by “lying in
” (one has to extend the Solymosi-de Zeeuw analysis to more general number fields, but this should be routine, using the analogue of Faltings’ theorem for such number fields).
— 1. Singularities —
Let us inspect the local behaviour of near an obvious singularity, when
is close to
for some
. We may normalise so that
and
, and then we may use the affine chart
, so that we are looking at the affine variety
for near
. Note that for
,
stays away from zero and so
is a smooth branch of the square root of
near an obvious singularity. Thus, up to an invertible analytic map, the local behaviour of
near the obvious singularity is that of a cone
Such a cone is blown up to the surface
where
is the closure of the equivalence class in
; the origin
blows up to the conic curve
and the blowup locally looks (on the level of real smooth manifolds, at least) like the cylinder . In particular, the blown up variety is smooth near the blowup of the original singularity.
Now we look at the behaviour of near a non-obvious singularity (5); for sake of discussion we work with the sign
. Here the calculations are messier; unfortunately, I do not know of a slick way to avoid excessive computation. We use the affine chart
and write
, then we are looking at the affine variety
for near
. We can rewrite this as
Using the equations, we can solve for
and
(using the general position of the
) to obtain equations of the form
for some homogeneous quadratics and complex coefficients
determined by the
. Substituting this into the
equations we obtain equations of the form
for some homogeneous quadratics and complex coefficients
determined by the
. One can solve the first two equations by power series (or the inverse function theorem) for
near zero to obtain an analytic representation
for some functions analytic near the origin with
. The second two equations then become
for some functions analytic near the origin that vanish to second order at
. Thus, near this non-obvious singularity,
is analytically equivalent to the complex surface
near .
It may be possible to simplify the surface further into an even better normal form, though I was unable to do so. Nevertheless, the current form is simple enough that one can understand the blowup (in the category of real smooth manifolds, at least), which in this case is
or equivalently the three-dimensional manifold
quotiented by the equivalence , where
is the function
which is locally analytic near the origin. One can cover this manifold by the four affine charts . For instance, the
chart becomes the affine manifold
The point blows up to the curve
that is to say the intersection of two quadrics in . For
generic, the vectors
and
are linearly independent, and one easily verifies that this is a smooth curve in
, which by Bezout’s theorem is of degree four. If one projects from a point of this curve to a generic hyperplane, one obtains a smooth planar curve of degree three, that is to say an elliptic curve
. Thus, on the level of real smooth manifolds at least, the blowup of
is equivalent to
, by the inverse function theorem; in particular, the blowup is smooth near the fibre of the original singularity.
To summarise, the blown up surface is smooth, with the obvious singularities blowing up to a conic curve
with neighbourhood structure
, and the non-obvious singularities blowing up to an elliptic curve
with neighbourhood structure
(at the level of real smooth manifolds; I was not able to understand the complex or algebrai geometry structure properly).
— 2. General type —
Let be a smooth complex projective variety of some dimension
. The canonical bundle
(also known as the determinant bundle) is then the top-dimensional exterior product of the cotangent bundle
, thus sections of this bundle are holomorphic
-forms on
. Since the space of possible top-dimensional forms at a point is one-dimensional, this is a line bundle. One can also take higher powers
of this bundle for any natural number
to create further line bundles, known as pluricanonical bundles; sections of the pluricanonical bundle of order
can then be locally represented as formal
-fold products of holomorphic
-forms.
For each pluricanonical bundle , define
to be the space of global sections of this bundle; in particular,
is the space of globally holomorphic
-forms. It turns out (as can be proven for instance using compactness theorems such as Montel’s theorem) that this space is always a finite-dimensional complex vector space; the global sections are also always algebraic (this follows from Serre’s GAGA paper, but presumably can be derived from earlier results also). If the space has some positive dimension
(that is, at least one non-trivial global section exists), we say that the bundle is effective, and then we can define an (almost everywhere defined) pluricanonical map
by the formula
where is a complex basis for the space
; note that this map can be undefined if the
all simultaneously vanish, but this is a measure zero set. The pluricanonical map is only defined up to choice of basis
, but changing the basis only amounts to applying a projective linear transformation
to
. In particular, the image
of
is well defined up to projective transformations. We refer to the
case
of the pluricanonical map (when it exists) as the canonical map.
The Kodaira dimension of is then defined to be the maximum of the dimensions of
for all natural numbers
with effective pluricanonical bundles, or
if no such
exists (some authors use
instead of
). From the definition it is clear that the Kodaira dimension of
is at most the dimension of
as an algebraic variety (or as a complex manifold). The variety
is said to be of general type if the two dimensions are equal, that is to say that there is a pluricanonical map
whose image has the same dimension as
.
Example 6 Suppose
is a projective space. On the one hand, an
-form on
is an object
that assigns to each point
an alternating
-form
from tangent vectors
to complex numbers. But
is the quotient of
by dilations. Thus, one can also view the
-form
as a lifted object
that assigns to each point
an alternating
-form
from complex vectors
to complex numbers, such that one has the vanishing property
if any of the
are a scalar multiple of
, as well as the scale invariance property
for any
, or equivalently
(This homogeneity of order
, together with some calculations involving the changes of coordinate between the different affine patches of projective space, can be used to show that the canonical bundle is isomorphic to the line bundle
, the
power of the tautological line bundle
.) In particular, if
is a globally holomorphic
-form,
are arbitrary vectors, and
is an arbitrary homogeneous polynomial of degree
, then
is a holomorphic map from
to
which is dilation invariant, and thus descends to a holomorphic function on
, which by Liouville’s theorem is then necessarily constant. By varying the polynomial
, this shows that
must vanish identically. Thus the canonical bundle here is not effective. A similar scaling argument shows that the pluricanonical bundles are not effective either, thus
has a Kodaira dimension of
(or
).
Example 7 In the case of smooth projective algebraic curves, it turns out that genus zero curves like
have Kodaira dimension
(or
), genus one curves (such as elliptic curves) have Kodaira dimension zero, and higher genus curves have Kodaira dimension one and are thus of general type. This is basically a corollary of the Riemann-Roch theorem. Intuitively, the more “holes” or other topologically interesting structure a variety
has, the more opportunity there is for the canonical and pluricanonical bundles to have interesting global sections, and the more likely it becomes that the variety is of general type.
Remark 8 As the name suggests, “most” varieties are expected to be of general type, with only a few “special” varieties being not of general type. In the case of curves, the special varieties are the genus zero and genus one curves. In the case of surfaces, the situation is analogous, but significantly more complicated, and is described by the Enriques-Kodaira classification. I don’t know the current level of understanding for higher dimensional varieties, but would imagine that the story becomes even more complicated than in the surface case.
— 3. Constructing differential forms —
To show that a smooth projective variety is of general type, it clearly suffices to locate enough global holomorphic
-forms that the canonical map has full dimension in its image. It is thus of interest to find ways to construct global holomorphic
-forms on such varieties.
Suppose first that we have a (possibly singular) complete intersection
of codimension in
, where
are homogeneous polynomials of degree
respectively. This variety can be viewed as the quotient of the quasiprojective variety
by dilations. Similarly to Example 6, an -form
on
can then be identified with an
-form
on
obeying the vanishing condition
whenever one of the is parallel to
, as well as the scaling relationship
To try to create such a form, we can start with the standard -form
on , and “divide” by the one-forms
to create a
-form
on
by requiring that
for ,
, and
, where
Away from the singularities of , the determinant form
is non-degenerate (after quotienting out the
by the tangent plane
, on which this form clearly vanishes). As such,
is well-defined as a form on the smooth points of
. It also obeys the vanishing condition (7) when one of the
is parallel to
. However, it doesn’t necessarily obey the scaling relationship (8); instead, one has
for any , where
is the quantity
Repeating the heuristic analysis in the introduction, we expect the number of integer points in of height
to be
, so
should morally correspond to “general type” in some sense. This is reflected here by the ability, when
, to multiply
by an arbitrary homogeneous polynomial
of degree
, leading to a form
which does obey both (7) and (8), and thus descends to an
-form defined on the smooth points of
. This already shows that for smooth complete intersections
, one has general type whenever
(because one can use
of the form
, for
a fixed homogeneous polynomial of degree
and
being the basis functions, to create a portion of the canonical map that is basically the tautological embedding of
to
). (Indeed, this argument shows in this case that the canonical bundle on
is isomorphic to the pullback of
to
.)
If has singularities, then the situation is a bit more complicated; we have to pass to the blowup of
, and the form
may develop some singularities as one approaches the blown up fibres. For the specific blowup
considered in Theorem 4, though, it turns out that these singularities are removable, if we make
vanish at the exceptional singular points (5); as there are only two such singular points, this will still give enough freedom in
to make the canonical map have full dimension in the image.
We turn to the details. Starting with the complete intersection given by (1), we let
be the
-form on the smooth portion of the deprojectivised variety
defined by the above construction, thus
for a smooth point of
,
, and
. In this case,
,
, and
, so
, and so for any linear function
, the form
descends to a
-form on the smooth portion of
, which then lifts to a
-form on the blown up variety
except possibly at the exceptional fibres above each of the singular points of
.
We now claim that this -form on
can be smoothly continued to each of the fibres if the linear function
vanishes at the two exceptional points (5), or equivalently that
is a linear combination of the coefficients
. Let us first verify this for an obvious singular point. As before, we normalise
and
, and take the affine chart
, then
is locally described by the affine variety
where as usual we identify with
, and the form
restricted to this affine variety takes the form
for ,
, and
. In particular, taking
for
, where
is the standard basis, we have
whenever the denominator is non-zero, where is the one-form on
that sends a tangent vector
to its
component
, and similarly for
,
, etc. Using the notation of wedge product
, we can write this as
Using (2), we compute that
The quantities are bounded away from zero near the singularity, and
is also bounded. Thus we have
near the singularity, where we write for
-forms
to denote the assertion that
for some scalar function that is bounded near the singularity.
At first glance, it looks like this form could blow up whenever
vanishes, but this does not actually happen as the numerator will also vanish in that case. One can see this by using a different choice of
for
in (9). For instance, if one takes
to be
rather than
, then we get
and now
so we obtain an alternate form
One can also see the equivalence of (10), (11), (12) by noting that vanishes on
, which on differentiating yields the linear dependence
between the -forms
, which can then easily be used to relate (10), (11), (12) to each other.
Thus the only actual singularity that could occur here is when all vanish. However, it turns out that even here the singularity is removable after blowing up the surface. Recall that the blowup surface
takes the form
We pick an affine chart of this surface; for sake of argument we take the chart, as the other charts
are treated similarly. Then the surface can be expressed as
with . In particular,
and thus
The key point is that there is a common factor of here. Using (11), we thus have
and so does not blow up in the coordinates of
as
approaches
. By Riemann’s theorem, (the pullback of)
can thus be extended holomorphically to the exceptional fibre
.
Now we look at the behaviour near a non-obvious singularity; for sake of argument we look at the neighbourhood of
As before, we use the affine chart and write
, then we are looking at the affine variety
near , where
is the system
of quadratic polynomials
Applying a change of variables, we have
for ,
, and
, and some harmless constant
(depending on
). As before, if we set
for
, we arrive at
when the denominator is non-zero. Similarly
when the denominators are non-zero. Similarly for permutations of the indices.
Now, we lift to the blowup manifold
For sake of discussion we work in the affine chart (say) and write
, in which case the manifold becomes
with the convention . To convert back into the original coordinates
, we have
and
Thus
and
and thus
Also, . From (14) we can thus cancel all the factors of
and conclude that
and so stays bounded up to the exceptional fibre in the blowup coordinates as long as
stay bounded away from zero.
It remains to deal with the case when one of the are small. From the form of the curve (6), and the fact that the
are in general position, none of the vectors
are scalar multiples of each other, which means that only one of the
can vanish at a time. Suppose
is close to vanishing, so that
are comparable to one (the other cases will be similar). In this case, we use (15). As before,
and
; also
and so
So once again the factors of cancel, and
and so stays bounded up to the exceptional fibre in this case. Similarly when one of the other
are small. Thus
stays bounded as one approaches the exceptional fibre in any direction, and so by Riemann’s theorem it can be continued holomorphically to this fibre.
In conclusion, lifts to a smooth
-form on the blowup surface
. By choosing
to be the coordinate functions
, we conclude that the image of the canonical map on
contains the image of
under the (generically defined) projection
. But it is easy to see that this map is generically injective on
(indeed, one can solve for
as rational functions of
), by subtracting the various equations
from each other), and so the image has full dimension. This establishes that
has general type as required.
22 comments
Comments feed for this article
20 December, 2014 at 3:06 pm
Sean Eberhard
Dear Terry,
This is really nice. I learned about the Erdos-Ulam problem recently, after I came across Harborth’s conjecture (1987): Every planar graph has a drawing in which all edges are straight line segments of integer length. Of course since the answer to the Erdos-Ulam problem is probably that such sets cannot be dense one must find a different approach to Harborth’s conjecture.
20 December, 2014 at 3:22 pm
Sean Eberhard
To further justify the connection with Harborth’s conjecture, I guess it seems likely that for
sufficiently large, say
or something, the set of
-tuples of points in
pairwise at rational distance is not dense in
. (I’m not sure whether your analysis gives this stronger variant.)
20 December, 2014 at 7:55 pm
Terence Tao
The analysis in the post shows that, given any four points in the plane at rational distances from each other and in general position, the set of points at rational distances from these original four lies in a finite union of plane algebraic curves and thus not dense. But that is weaker than the claim above. In principle, though, the set of n-tuples with this property is the set of rational points of some variety, and for
large enough it is likely that this variety becomes of general type (after blowing up to a smooth projective variety). But the computations are likely to be much more complicated than those in this post, especially given that the variety looks to have a fairly large dimension (something like
).
20 December, 2014 at 8:03 pm
Fan
“roughly speaking, it one of the most optimistic natural conjecture one could make that is compatible with these heuristics while also being invariant under birational equivalence” missing “is” after “it”?
[Corrected, thanks – T.]
21 December, 2014 at 1:07 pm
John Mangual
I don’t know much algebraic geometry. Janos Kollar likes to mix asymptotic in there such as this paper http://arxiv.org/abs/1405.2243
23 December, 2014 at 12:37 am
Jafar Shaffaf
Dear Terry,
I didn’t know that you are interested in Ulam conjecture, I proved Ulam conjeture last year but I didn’t publish it yet. In fact I associate a special surface in $\mathbb{P}^3$ to a rational subset of the plane and I call them distance surface. I prove that these surfaces are surfaces of general type. For more detail I send this paper to your email and I hope you can see this and I appreciate it if you have any comment/idea about that.
26 December, 2014 at 1:32 pm
Anonymous
Since it is announced on a blog, I suppose there is no secrecy about it: Terry, can you (dis)confirm that you have indeed received an email from Shaffaf with a claimed proof of the Ulam conjecture?
[Yes, I have seen the manuscript, but (as with this post) the proof is conditional on the Bombieri-Lang conjecture. -T.]
23 December, 2014 at 8:54 am
jozsef
Very nice! Do you have a uniform bound on the number of curves (and their degree) covering the points at rational distances from the four points? It seems that your argument implies a stronger statement. If we suppose Bombieri-Lang then there is an N such that any planar integral pointset in general position has at most N elements. (integral pointset = every distance is integer). The best known lower bound is N=7 by a construction of Sascha Kurz.
23 December, 2014 at 9:48 am
Terence Tao
Well, one could simply add such a uniformity claim to the Bombieri-Lang conjecture one is using as input, but this is of course cheating. There is a paper of Caporaso, Harris, and Mazur that uses Bombieri-Lang (in its original form) to deduce a uniform version, but only for curves (i.e. Faltings’ theorem). (I learned of this reference from the paper of Makhul and Shaffaf referenced in the post.) I don’t know if there is any hope of extending these arguments to surfaces.
I think I agree though that if such a uniformity claim was present, then by combining the arguments here with those in your paper with Frank one should be able to get a fixed upper bound N on rational-distance points in general position (though one may need a strong version of general position in which one avoids more configurations than collinear points or quadruples on a circle).
28 December, 2014 at 9:21 pm
Richard
Re: “PS: Please take down the rating system.”
Just Adblock everything from poll.fm: rule “||poll.fm^”
I did that to kill off the up/down comment votes business long ago, not just because they aren’t useful (and they aren’t at all useful), but because contacting poll.fm can lead to very lengthy delays in loading simple wordpress pages.
25 December, 2014 at 5:30 am
Teresa
dear proffessor tao,
I m really amazing of your math intelligence. i want to know if you ever solved a conjecture, or if is a problem who you can t solve. wait for your answer, sorry for my language i dont know very well english language. happy for you, wait a answer..
26 December, 2014 at 2:34 am
sonam gupta
Hi,
That was nicely written. I really enjoyed your post.
Thanks to share this post:)
Regards,
Sonam Gupta
28 December, 2014 at 7:42 pm
Anonymous
Dear Terry,
Suddenly, I discover a valuable secrect:among Field medalist or Nobelist never cooperate with each other.Simple reason, among two kings never listen to each other’s idea.Another reason, all for selfish and envy.Among them , everyone thinks that he/she is the best.Once again, Yin-Yang princple of China is always true, forever.From now to 2015, 2050., …2900, no matter how good sience is , human beings can not kill their selfish or “nature of myself according to Taoism”.There is not two suns in our universe? If Field medalist or Nobelist humble each other, I’m sure we can solve 100 best hard problems or discovering many medical treatmnents for human.I’m nothing, i’m not the best intelligent man.But I think nowadays,the advance of sience is nothing.We cannot call” Paradise of sience”.For example, many diseases cannot be treated by sience , sometime olny one small leaf in nature can do very well.
29 December, 2014 at 9:20 am
How many rational distances can there be between N points in the plane? | Quomodocumque
[…] has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me. Here’s the […]
29 December, 2014 at 9:24 am
JSE
I like this story! Idle question, from my blog: is there a “quantitative Erdos-Ulam problem?” That is: if S is a set of N points in the plane of which no more than M lie on any line or conic, how many of the ~N^2 distances between points of S can be rational? You can easily get NM; is it easy to get much more? https://quomodocumque.wordpress.com/2014/12/29/how-many-rational-distances-can-there-be-between-n-points-in-the-plane/
3 January, 2015 at 10:35 am
Jafar Shaffaf
I sent the paper to arxiv, it might be useful and interesting for someone.
3 January, 2015 at 10:32 pm
Janko Bračič
A very nice post! However, I have a question: it is said in the beginning “(Conversely, it is easy to construct examples of dense subsets of a line or circle in which all distances are rational.) ” I guess that this does not hold for any circle. It seems that the square of the radius of the circle has to be rational number.
[Yes; clarification added, thanks – T.]
13 March, 2015 at 7:21 am
Elena Citro
Hi ! I apologize in advance for the futility of my message ( in comparison to the beauty of the topics covered ) , but it’s for a few years that I have a curiosity about your talent and did not know where to send it . I know that many prodigies are synesthetes , you also have the gift of synesthesia ?
29 July, 2015 at 3:57 am
Georges Elencwajg
Dear Terence, in your boxed example 6 you seem to imply that the canonical bundle of $\mathbb P^n$ is $\mathcal O(-n)$, while in reality it is $\mathcal O(-n-1)$. What am I misunderstanding ?
[Oops, I had not done the change of coordinate calculations carefully enough; the comment is now corrected. -T.]
26 November, 2018 at 12:20 pm
Terence Tao
Sándor Kovács has kindly written up a proof of Theorem 4 in more conventional algebraic geometry language, using the language of sheaves and linear systems. The main point is that as the singularities of V are all either canonical or log-canonical, one can relate sheaves on V with sheaves on the resolution in a relatively straightforward manner, so that general type can be verified more or less directly on V rather than on the resolution. I have uploaded his notes at https://terrytao.files.wordpress.com/2018/11/bl-to-eu.pdf .
8 June, 2019 at 8:01 am
Ruling out polynomial bijections over the rationals via Bombieri-Lang? | What's new
[…] a powerful conjecture in Diophantine geometry known as the Bombieri-Lang conjecture (discussed in this previous post), it is at least possible to exhibit polynomials which are […]
10 December, 2020 at 2:34 pm
What if they are all wrong? | Igor Pak's blog
[…] since it would both imply one conjecture and give a counterexample to another? For the record, I wouldn’t do that, just making a polemical […]