Related papers: The average amount of information lost in multipli…

A fundamental problem in data management is to find the elements in an array that match a query. Recently, learned indexes are being extensively used to solve this problem, where they learn a model to predict the location of the items in…

Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup_n (n\log\log n)^{-1}|\sum_{1<=…

The mutual information between a set of stimuli and the elicited neural responses is compared to the corresponding decoded information. The decoding procedure is presented as an artificial distortion of the joint probabilities between…

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently…

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and…

We find a searching method on ordered lists that surprisingly outperforms binary searching with respect to average query complexity while retaining minmax optimality. The method is shown to require $O(\log_2\log_2 n)$ queries on average…

The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_XP_Y(E)$ (where $P_XP_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$.…

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log…

Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $\theta_i$ and scale parameter $\lambda_i$ (resp.…

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…

It is well understood that if one is given a set $X \subset [0,1]$ of $n$ independent uniformly distributed random variables, then $$ \sup_{0 \leq x \leq 1} \left| \frac{\# X \cap [0,x]}{\# X} - x \right| \lesssim \frac{\sqrt{\log{n}}}{…

Let $\Phi(x,y)$ denote the number of integers $n\in[1,x]$ free of prime factors $\le y$. We show that but for a few small cases, $\Phi(x,y)<.6x/\log y$ when $y\le\sqrt{x}$.

An integer is said to be $y$-friable if its greatest prime factor P(n) is less than $y$. In this paper, we study numbers of the shape $n-1$ when $P(n)\leq y$ and $n\leq x$. One expects that, statistically, their multiplicative behaviour…

This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ are i.i.d., and $a_n$ and $b_n$ are independent with $P(a_n=2)=P(a_n=(p+1)/2)=1/2$ and…

Hemaspaandra, Hempel, and Wechsung [cs.CC/9909020] raised the following questions: If one is allowed one question to each of two different information sources, does the order in which one asks the questions affect the class of problems that…

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log…

We consider the uniform distribution of solutions $(x,y)$ to $xy=N \mod a$, and obtain a bound on the second moment of the number of solutions in squares of length approximately $a^{1/2}$. We use this to study a new factoring algorithm that…

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…

The combined Universal Probability M(D) of strings x in sets D is close to max M({x}) over x in D: their ~logs differ by at most D's information j=I(D:H) about the halting sequence H. Thus if all x have complexity K(x) >k, D carries >i bits…

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main…