Sequences of integers with missing differences

This paper deals with the following problem posed by Professor T. S. Motzkin: Suppose M is a given set of positive integers. How dense can a set S of positive integers be, if no two elements of S are allowed to differ by an element of M? The problem is solved for |M| ? 2, and some partial results are obtained in the general case.     

Sets characterized by missing sums and differences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We show that the probability that a uniform random subset of {0,1,…,n} is an MSTD set approaches some limit ρ>4.28×¹⁰−4. This improves the previous result of Martin and O?Bryant that there is a lower limit of at least 2×¹⁰−7. Monte Carlo experiments suggest that ρ≈4.5×¹⁰−4. We present a deterministic algorithm that can compute ρ up to arbitrary precision. We also describe the structure of a random MSTD set S⊆{0,1,…,n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any “middle” element is in S approaches 1/2 as n→∞, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

Convergent series of integers with missing digits

The Ramanujan Journal - A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger...     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

Cycles of differences of integers

Let A = (a_j) be a k-tuple of positive integers. We define a new k-tuple $\text{A}\text{= (|a}{}_{\mathrm{j}}\text{? a}{}_{\mathrm{j\; +\; 1}}\text{|)}$ by taking numerical differences. If this process is repeated, eventually repetition takes place, resulting in a cycle. We show that except for constant multiples there are only a finite number of cycles. We determine explicitly those k-tuples which are in a cycle.     

Sequences of integers with missing quotients and dense points without neighbors

Let $A$ be a pre-defined set of rational numbers. We say that a set of natural numbers $S$ is an $A$-quotient-free set if no ratio of two elements in $S$ belongs to $A$. We find the maximal asymptotic density and the maximal upper asymptotic density of $A$-quotient-free sets when $A$ belongs to a particular class.It is known that in the case $A=\{p,q\}$, where $p$, $q$ are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.     

Arithmetic properties of integers with missing digits: distribution in residue classes

We show that numbers with missing digits are in average well-distributed in residue classes mod m where averaging is taken over m.     

On arithmetic properties of integers with missing digits II: Prime factors

Prime factors of numbers with missing digits are studied. It is shown that, under certain conditions, this set satisfies an Erd s-Kac type theorem; it contains numbers with ‘many’ prime factors; it contains numbers whose greatest prime factor is ‘large’ (resp. ‘small’).     

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums.      

On differences and sums of integers,I

A set {b₁,b₂,…,b_i} ? {1,2,…,N} is said to be a difference intersector set if {a₁,a₂,…,a_s} ? {1,2,…,N}, j > ?N imply the solvability of the equation a_x ? a_y = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b₁,b₂,…,b_i} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations ($\text{a}{}_{\mathrm{x}}\text{?}\text{a}{}_{\mathrm{y}}\text{p}\text{= + 1}$, $\text{(a}{}_{\mathrm{u}}\text{?}\text{a}{}_{\mathrm{v}}\text{p}\text{) = ? 1}$, $\text{(a}{}_{\mathrm{r}}\text{+}\text{a}{}_{\mathrm{s}}\text{p}\text{) = + 1}$, $\text{(a}{}_{\mathrm{t}}\text{+}\text{a}{}_{\mathrm{z}}\text{p}\text{) = ? 1}$ (where ($\text{a}\text{p}$) denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets.     

Representing integers as sums or differences of general power products

We extend a result of Hajdu and Tijdeman concerning the smallest number which cannot be obtained as a sum of less than k power products of fixed primes. For this, we also generalize a classical result of Tijdeman concerning the gaps between integers of the form $p_{1}^{\alpha_{1}}\cdots p_{t}^{\alpha_{t}}$ where the p _i are primes, to the case where the numbers p _i are not necessarily primes.     

Empirical likelihood confidence intervals for the differences of quantiles with missing data

Suppose that there are two nonparametric populations x and y with missing data on both of them. We are interested in constructing confidence intervals on the quantile differences of x and y. Random imputation is used. Empirical likelihood confidence intervals on the differences are constructed. Supported by the National Natural Science Foundation of China (No. 10661003) and Natural Science Foundation of Guangxi (No. 0728092).     

Semi-empirical likelihood confidence intervals for the differences of quantiles with missing data

Detecting population （group） differences is useful in many applications, such as medical research. In this paper, we explore the probabilistic theory for identifying the quantile differences .between two populations. Suppose that there are two populations x and y with missing data on both of them, where x is nonparametric and y is parametric. We are interested in constructing confidence intervals on the quantile differences of x and y. Random hot deck imputation is used to fill in missing data. Semi-empirical likelihood confidence intervals on the differences are constructed.     

On the density of integral sets with missing differences from sets related to arithmetic progressions

For a given set M of positive integers, a problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by an element of M. The problem is completely settled when |M|?2, and some partial results are known for several families of M for |M|?3, including the case where the elements of M are in arithmetic progression. We consider some cases when M either contains an arithmetic progression or is contained in an arithmetic progression.     

Fibonacci integers

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77=21⋅55/(3⋅5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them is between and , for an explicitly determined constant c. The proof is based on both combinatorial and analytic arguments.