Related papers: Random sparse unary predicates

A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\ell(k)$ be the smallest integer $\ell$ such that any…

The classical Poisson theorem says that if $\xi_1,\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^n\xi_i$ is asymptotically in $n$ Poisson distributed with…

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the…

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances…

The Gauss--Lucas theorem states that any convex set $K\subset\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\in\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following…

Let $p \in (0,1/2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t \ge 0$, \[\mathbb{P}[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n} + C_p…

In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.

Consider an election between k candidates in which each voter votes randomly (but not necessarily independently) and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for…

Although squaring integers is deterministic, squares modulo a prime, $p$, appear to be random. First, because they are all generated by the multiplicative linear congruential equation, $x_{i+1} = g^2 x_i \mod p$, where $x_0 = 1$ and $g$ is…

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

A linear equation $E$ is said to be sparse if there is $c>0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterizing the sparse equations, first raised by Ruzsa in the…

Given positive integers $n$ and $m$, let $p_n(m)$ be the probability that a uniform random permutation of $[n]$ has order exactly $m$. We show that, as $n \to \infty$, the maximum of $p_n(m)$ over all $m$ is asymptotic to $1/n$, the…

Sorting is a fundamental problem in computer science. In the classical setting, it is well-known that $(1\pm o(1)) n\log_2 n$ comparisons are both necessary and sufficient to sort a list of $n$ elements. In this paper, we study the Noisy…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a…

Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $\zeta_1,\ldots,\zeta_n$ denote the roots of $f$. We show that the point…

Zero-one laws state that probabilistic events of a certain type must occur with probability either $0$ or $1$, and nothing in between. We formulate a syntactic zero-one law, which enjoys good logical properties while being broadly…