Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used byQuicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16.

Consider the domain Z " = fx 2 Rn j dist(x; "Zn) ? "flg; and let the free path length be defined as

Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy

Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an r-th power residue for all small factors of p − 1. The corresponding Diffie-Hellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen

Abstract. We will show that the famous, intractible 1959 Kadison-Singer problem in C ∗-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer. 1.

. We are concerned with the problem of minimizing the supremum norm on an interval of a non-zero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any non-trivial cases. We examine the case of the interval [0; 1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal "integer Chebyshev" polynomials, showing for example, that on small intevals [0; ffi] and for small degrees d, x d achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys' and others, as to what the "integer transfinite diameter" of [0; 1] should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due t...

The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique xed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric. AMS 2000 subject classications. Primary 68W40; secondary 68P10, 60E05, 60E10, 60F05. Key words and phrases. Quicksort, characteristic function, density, moment generating function, sorting algorithm, coupling, Fourier analysis, Kolmogorv{Smirnov distance, total variation distance, integral equation, numerical analysis, d p -metric. Date. January 15, 2001; modied August 22, 2001. 1 Research supported by NSF grant DMS{9803780, and by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. 1 1

Abstract. We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erdős, which says that the number of distances determined by n points in Rd is at least Cdn 1 d +ǫd, ǫd> 0. Introduction and statement of results Fourier bases. Let D be a domain in R d, i.e., D is a Lebesgue measurable subset of R d with finite non-zero Lebesgue measure. We say that D is a spectral set if L 2 (D) has orthogonal basis of the form EΛ = {e 2πix·λ}

A central problem in analytic number theory is to gain an understanding of character sums χ(n), n≤x

: We prove several results related to a question of Steinhaus: is there a set E ae R 2 such that the image of E under each rigid motion of R 2 contains exactly one lattice point? Assuming measurability we answer the analogous question in higher dimensions in the negative, and we improve on the known partial results in the two dimensional case. We also consider a related problem involving finite sets of rotations. The following question was raised by Steinhaus in 1957 and has been the subject of several recent papers. Does there exist a set E ae R 2 such that every rotation and translation of E contains exactly one integer lattice point? By a rotation and translation of a set E ae R d we mean of course a set of the form aeE+x for some ae 2 SO(d) and x 2 R d . It is natural to consider Steinhaus' question separately for measurable and nonmeasurable sets. Both the measurable and nonmeasurable cases are presently open, but this paper will be concerned only with the measura...