empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

arithmetic differential equations on gln i differential cocycles abstract the theory of differential equations has an arithmetic analogue 8 in which derivatives are replaced by fermat quotients one can then ask what is the arithmetic analogue of a linear differential equation the study of usual linear differential equations is the same as the study of the differential cocycle from gl n into its lie algebra given by the logarithmic derivative 14 equivalently by the maurer cartan connection 17 however we prove here that there are no such cocycles in the context of arithmetic differential equations in sequels of this paper 10 11 we will remedy the situation by introducing arithmetic analogues of lie algebras and a skew version of differential cocycles this will lead to a theory of linear arithmetic differential equations

on decentralized estimation with active queries we consider the problem of decentralized 20 questions with noise for multiple players agents under the minimum entropy criterion in the setting of stochastic search over a parameter space with application to target localization we propose decentralized extensions of the active query based stochastic search strategy that combines elements from the 20 questions approach and social learning we prove convergence to correct consensus on the value of the parameter this framework provides a flexible and tractable mathematical model for decentralized parameter estimation systems based on active querying we illustrate the effectiveness and robustness of the proposed decentralized collaborative 20 questions algorithm for random network topologies with information sharing

characterizations of l n and l1n and their stabilities we characterize l n and l1n by introducing two geometric constants in n dimensional banach spaces we also prove that these characterizations are stable with respect to the banach mazur distance

effective bounds for the number of transcendental points on subvarieties of semi abelian varieties let a be a semi abelian variety and x a subvariety of a both denned over a number field assume that x does not contain x x2 for any positive dimensional subvarieties x x2 of a let r be a subgroup of a c of finite rational rank we give doubly exponential bounds for the size of x n 0 x q among the ingredients is a uniform bound doubly exponential in the data on finite sets which are quantifier free definable in differentially closed fields we also give uniform bounds on x n v in the case where x contains no translate of any semi abelian subvariety of a and r is a subgroup of a c of finite rational rank which has trivial intersection with a q here a is assumed to be defined over a number field but x need not be algebraic torus c 2 and both a and x are defined over a number field the question asked by bombieri was to find uniform bounds for the number of generic points of x fl t namely those points which are not rational over the al gebraic closure of q buium obtained uniform bounds of an iterated exponential nature in the data deg x 9 rank t buium used differential algebraic methods the aim in the present paper is to improve the bounds to being doubly exponen tial in the relevant data as well as to treat the more general case of semi abelian varieties we are also able to slightly weaken the hypothesis on x the methods are also differential algebraic closely following buium but there are additional model theoretic ingredients coming from the theory of commutative groups of finite morley rank and there is also one more key observation regarding the differential algebraic set up which yields the doubly exponential bounds before stating the results let us discuss the nature of the data which for us will be quite naive we work for now over the complex numbers let a be a semi abelian variety of dimension n9 and x a subvariety of dimension ra both defined over q denote q by k assume a to be given as a locally closed

almost euclidean sections in symmetric spaces and concentration of order statistics abstract we prove that for any symmetric n dimensional normed space e and any e 0 e contains a 1 e euclidean subspace of dimension at least c ln n ln 1 e where c is an absolute constant the proof is based on a concentration property of order statistics of random vectors with i i d coordinates

collaborative 20 questions for target localization we consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search with application to target localization each player yields a noisy response to a binary query governed by a certain error probability first we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location second we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions this generalizes the single player probabilistic bisection method for stochastic search problems third we prove an equivalence between the two schemes showing that despite the fact that the sequential scheme has access to a more refined filtration the joint scheme performs just as well on average fourth we establish convergence rates of the mean square error and derive error exponents finally we obtain an extension to the case of unknown error probabilities this framework provides a mathematical model for incorporating a human in the loop for active machine learning systems

dichotomies structure and concentration in normed spaces abstract we use probabilistic topological and combinatorial methods to establish the following deviation inequality for any normed space x r n there exists an invertible linear map t r n r n with p t g e t g e e t g c exp c max e 2 e log n e 0 where g is the standard n dimensional gaussian vector and c c 0 are universal constants it follows that for every e 0 1 and for every normed space x r n there exists a k dimensional subspace of x which is 1 e euclidean and k c e log n log 1 e this improves by a logarithmic on e term the best previously known result due to g schechtman

on dvoretzky s theorem for subspaces of lp abstract we prove that for any 2 p and for every n dimensional subspace x of l p represented on r n whose unit ball b x is in lewis position one has the following two level gaussian concentration inequality p z e z e e z c exp c min p e 2 n e n 2 p 0 e 1 where z is the standard n dimensional gaussian vector p 0 is a constant depending only on p and c c 0 are absolute constants as a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces in particular for any 2 p and every n dimensional subspace x of l p the euclidean space l 2 k can be 1 e embedded into x with k c p min e 2 n e n 2 p where c p 0 is a constant depending only on p this improves upon the previously known estimate due to figiel lindenstrauss and milman

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

explicit euclidean embeddings in permutation invariant normed spaces abstract let x be a real normed space of dimension n n with a basis e i 1 n such that the norm is invariant under coordinate permutations assume for simplicity that the basis constant is at most 2 consider any n n and 0 e 1 4 such that n c log e 1 1 log n we provide an explicit construction of a matrix that generates a 1 e embedding of l 2 n into x

metric and classical fidelity uncertainty relations for random unitary matrices we analyze uncertainty relations on finite dimensional hilbert spaces expressed in terms of classical fidelity which are stronger than metric uncertainty relations introduced by fawzi hayden and sen we establish the validity of fidelity uncertainty relations for random unitary matrices with optimal parameters up to universal constants which improves upon known results for the weaker notion of metric uncertainty this result is then applied to locking classical information in quantum states and allows to obtain optimal locking in hellinger distance improving upon previous results on locking in the total variation distance both by strengthening the metric used and by improving the dependence on parameters we also show that general probabilistic estimates behind the main theorem can be used to prove existence of data hiding schemes with bayesian type guarantees as a byproduct of our approach we obtain existence of almost euclidean subspaces of the matrix spaces with a better dimension distortion dependence than allowed in previously known constructions

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

isocrystals associated to arithmetic jet spaces of abelian schemes abstract using buium s theory of arithmetic differential characters we construct a filtered f isocrystal h a k associated to an abelian scheme a over a p adically complete discrete valuation ring with perfect residue field as a filtered vector space h a k admits a natural map to the usual de rham cohomology of a but the frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one when a is an elliptic curve we show that h a k has a natural integral model h a which implies an integral refinement of a result of buium s on arithmetic differential characters the weak admissibility of h a k depends on the invertibility of an arithmetic differential modular parameter thus the fontaine functor associates to suitably generic a a local galois representation of an apparently new kind

corrections to on decentralized estimation with active queries we provide a counterexample to a key lemma used in the proofs of the convergence of decentralized estimation algorithms in 2 we also provide an alternative lemma that establishes a new proof of the convergence results in the paper 2

the basic geometry of witt vectors ii spaces this is an account of the algebraic geometry of witt vectors and arithmetic jet spaces the usual p typical witt vectors of p adic schemes of finite type are already reasonably well understood the main point here is to generalize this theory in two ways we allow not just p typical witt vectors but those taken with respect to any set of primes in any ring of integers in any global field for example this includes the big witt vectors we also allow not just p adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field we give similar generalizations of buium s formal arithmetic jet functor which is dual to the witt functor we also give concrete geometric descriptions of witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors

ultrametric subsets with large hausdorff dimension it is shown that for every e 0 1 every compact metric space x d has a compact subset s x that embeds into an ultrametric space with distortion o 1 e and dim h s geqslant 1 varepsilon dim h x where dimh denotes hausdorff dimension the above o 1 e distortion estimate is shown to be sharp via a construction based on sequences of expander graphs

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

the mordell lang conjecture for function fields we give a proof of the geometric mordell lang conjecture in any characteristic our method involves a model theoretic analysis of the kernel of manin s homomorphism and of a certain analog in characteristic p department of mathematics massachusetts institute of technology 2 277 cambridge massachusetts 02139 current address department of mathematics hebrew university jerusalem israel e mail address ehud math mit edu

rational points of varieties with ample cotangent bundle over function fields let k be the function field of a smooth curve over an algebraically closed field k let x be a scheme which is smooth and projective over k suppose that the cotangent bundle omega x k is ample let r mathrm zar x k cap x be the zariski closure of the set of all k rational points of x endowed with its reduced induced structure we prove that for each irreducible component mathfrak r of r there is a projective variety mathfrak r 0 over k and a finite and surjective k mathrm sep morphism mathfrak r 0 k mathrm sep rightarrow mathfrak r k mathrm sep which is birational when mathrm char k 0 this improves on results of noguchi and martin deschamps in characteristic 0 in positive characteristic our result can be used to give the first examples of varieties which are not embeddable in abelian varieties and satisfy an analog of the bombieri lang conjecture

geometry of differential polynomial functions ii algebraic curves introduction the present paper is a direct continuation of b5 we shall freely borrow terminology and notations from that paper which shall be referred to from now on as part i however understanding the introduction of the present paper only requires familiarity with the introduction and first appendix of part i here we investigate 6 polynomial functions of smooth projective curves x defined over a 6 closed field f we will prove the surprising result that any smooth projective curve of genus 2 over f which does not descend to c can be naturally embedded via a 6 regular map as a 6 closed subset of a projective space which lies completely outside a unique hyperplane this provides in particular remarkable new examples of ritt s anomaly on the 6 dimension of intersections r p 133 we will also prove an infinitesimal analogue of serge lang s conjecture la which in special significant cases implies our 6 algebraic analogue of lang s conjecture b4 and hence the geometric analogue of lang s conjecture let f be as in the introduction of part i a 6 closed field with derivation operator 6 and constant field c and let x be a smooth projective f variety then one can always define two sequences of 6 regular maps

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

superconcentration and randomized dvoretzky s theorem for spaces with 1 unconditional bases abstract let n be a sufficiently large natural number and let b be an origin symmetric convex body in r n in the l position and such that the space r n b admits a 1 unconditional basis then for any e 0 1 2 and for random c e log n log 1 e dimensional subspace e distributed according to the rotation invariant haar measure the section b e is 1 e euclidean with probability close to one this shows that the worst case dependence on e in the randomized dvoretzky theorem in the l position is significantly better than in john s position it is a previously unexplored feature which has strong connections with the concept of superconcentration introduced by s chatterjee in fact our main result follows from the next theorem let b be as before and assume additionally that b has a smooth boundary and e n b n c e n grad b 2 for a small universal constant c 0 where grad b is the gradient of b and n is the standard gaussian measure in r n then for any p 1 c log n the p th power of the norm b p is c log n superconcentrated in the gauss space

arithmetic partial differential equations ii abstract we continue the study of arithmetic partial differential equations initiated in 7 by classifying arithmetic convection equations on modular curves and by describing their space of solutions certain of these solutions involve the fourier expansions of the eisenstein modular forms of weight 4 and 6 while others involve the serre tate expansions mori 1995 13 buium 2003 4 of the same modular forms in this sense our arithmetic convection equations can be seen as unifying the two types of expansions the theory can be generalized to one of arithmetic heat equations on modular curves but we prove that they do not carry arithmetic wave equations finally we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation

small width low distortions quantized random embeddings of low complexity sets under which conditions and with which distortions can we preserve the pairwise distances of low complexity vectors e g for structured sets such as the set of sparse vectors or the one of low rank matrices when these are mapped or embedded in a finite set of vectors this work addresses this general question through the specific use of a quantized and dithered random linear mapping which combines in the following order a sub gaussian random projection in mathbb r m of vectors in mathbb r n a random translation or dither of the projected vectors and a uniform scalar quantizer of resolution delta 0 applied componentwise thanks to this quantized mapping we are first able to show that with high probability an embedding of a bounded set mathcal k subset mathbb r n in delta mathbb z m can be achieved when distances in the quantized and in the original domains are measured with the ell 1 and ell 2 norm respectively and provided the number of quantized observations m is large before the square of the gaussian mean width of mathcal k in this case we show that the embedding is actually quasi isometric and only suffers from both multiplicative and additive distortions whose magnitudes decrease as m 1 5 for general sets and as m 1 2 for structured set when m increases second when one is only interested in characterizing the maximal distance separating two elements of mathcal k mapped to the same quantized vector i e the consistency width of the mapping we show that for a similar number of measurements and with high probability this width decays as m 1 4 for general sets and as 1 m for structured ones when m increases finally as an important aspect of this paper we also establish how the non gaussianity of sub gaussian random projections inserted in the quantized mapping e g for bernoulli random matrices impacts the class of vectors that can be embedded or whose consistency width provably decays when m increases