The structure of d-dimensional sets with small sumset

We describe the structure of d-dimensional sets of lattice points, having a small doubling property. Let K be a finite subset of Zd such that dimK=d?2. If and |K|>3⋅d⁴, then K lies on d parallel lines. Moreover, for every d-dimensional finite set K⊆Zd that lies on d?1 parallel lines, if , then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than terms. These best possible results answer a recent question posed by Freiman and cannot be sharpened by reducing the quantity v or by increasing the upper bounds for |K+K|.     

Small generators of number fields

If K is a number field of degree n over Q with discriminant D _K and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d ² such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real number field of prime degree n then one can find an integral generator α with . Received: 10 January 1997 / Revised version: 13 January 1998     

A Lower Bound for the Translative Kissing Numbers of Simplices

H (K) of a d-dimensional convex body K is the maximum number of mutually non-overlapping translates of K that can be arranged so that all touch K. In this paper we show that holds for any d-dimensional simplex (). We also prove similar inequalities for some, more general classes of convex bodies. Received May 18, 1998     

On the Discrepancy for Boxes and Polytopes

 We consider so-called Tusnády’s problem in dimension d: Given an n-point set P in R ^d , color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in differs from the number of blue points in by at most Δ, where should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that , with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R ^d . Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves and simplifies results of Beck and of Károlyi. Received 17 November 1997; in revised form 30 July 1998     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

Algebraic points of small height missing a union of varieties

Text

Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN, N?2. Let ZK be a union of varieties defined over K such that V?ZK. We prove the existence of a point of small height in V?ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing ZK, where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) [6] and [7]. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma (Thunder, 1995) [21] to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R-o6lr8s0Go.     

A lattice point problem and additive number theory

For every dimensiond1 there exists a constantc=c(d) such that for alln1, every set of at leastcn lattice points in thed-dimensional Euclidean space contains a subset of cardinality preciselyn whose centroid is also a lattice point. The proof combines techniques from additive number theory with results about the expansion properties of Cayley graphs with given eigenvalues.     

Some remarks concerning kissing numbers,blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3 ^d –1.     

A note on an inverse problem for lattice points

Let K⊆R² be a compact set such that K+Z²=R² and let K−K={a−b|a,b∈K}. We prove, via Algebraic Topology, that the integer points of the difference set of K, (K−K)∩Z², is not contained on the coordinate axes, Z×{0}∪{0}×Z. This result implies the negative answer to the inverse problem posed by M.B. Nathanson (2010) [5].     

Strong Version of the Basic Deciding Algorithm for the Existential Theory of Real Fields

Let U be a real algebraic variety in the n-dimensional affine space that is a set of all zeros of a family of polynomials of degree less than d. In the case where U is bounded (this is the main case), an algorithm of polynomial complexity is described for constructing a subset of U with the number of elements bounded from above by dⁿ that has the following property: for every s, this set has a nonempty intersection with every d-dimensional cycle with coefficients from s of the closure of the set of smooth points of dimension s of U. Bibliography: 16 titles.     

An application of an optimal behaviour of the greedy solution in number theory

Leta ₁,a ₂, ...,a _n be relative prime positive integers. The Frobenius problem is to determine the greatest integer not belonging to the set { _j=1 ⁿ a _j x _j :xZ ₊ ⁿ }. The Frobenius problem belongs to the combinatorial number theory, which is very rich in methods. In this paper the Frobenius problem is handled by integer programming which is a new tool in this field. Some new upper bounds and exact solutions of subproblems are provided. A lot of earlier results obtained with very different methods can be discussed in a unified way.     

Some characterisations of the ellipsoid

We show that a convex bodyK of dimensiond≧3 is an ellipsoid if it has any of the following properties: (1) the “grazes” of all points close toK are flat, (2) all sections of small diameter are centrally symmetric, (3) parallel (d−1)-sections close to the boundary are width-equivalent, (4)K is strictly convex and all (d−1)-sections close to the boundary are centrally symmetric; the last two results are deduced from their 3-dimensional cases which were proved by Aitchison.     

On Translational Clouds for a Convex Body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound . Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show . Finally, for the d-dimensional ball B^d we obtain the bounds .     

On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K⊂Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by d² smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K⊂Rd one has

     

Dual vectors and lower bounds for the nearest lattice point problem

We prove that given a point outside a given latticeL then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To be more precise, there is a set of translated hyperplanesH _i, such thatL _iH_i andd( _iH_i)(6n ²+1)^–1 d( ,L).Supported by an IBM fellowship.     

Exponential Lower Bound for the Translative Kissing Numbers of d -Dimensional Convex Bodies

The translative kissing number H(K) of a d -dimensional convex body K is the maximum number of mutually nonoverlapping translates of K which touch K . In this paper we show that there exists an absolute constant c > 0 such that H(K)≥ 2 ^cd for every positive integer d and every d -dimensional convex body K . We also prove a generalization of this result for pairs of centrally symmetric convex bodies. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p447.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received February 18, 1997, and in revised form April 15, 1997.     

On a spherical integral transformation and sections of star bodies

LetK be ad-dimensional star body (with respect to the origino). It is known that the (d–1)-dimensional volume of the intersections ofK with the hyperplanes througho does not uniquely determineK. Uniqueness can only be achieved under additional assumptions, such as central symmetry. Here it is pointed out that if one uses, instead of intersections by hyperplanes, intersections by half-planes that containo on the boundary, then, without any additional assumptions, the volume of these intersections determinesK uniquely. This assertion, and more general results of this kind, together with stability estimates, are obtained from uniqueness results and estimates concerning a particular spherical integral transformation.Supported by National Science Foundation Research Grant DMS-9401487     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005