Related papers: On a problem involving unit fractions

We prove that a subset $A\subseteq [1, N]$ with \[\sum_{n\in A}\frac{1}{n} \ge (\log N)^{4/5 + o(1)}\] contains a subset $B$ such that \[\sum_{n\in B} \frac{1}{n} = 1.\] Our techniques refine those of Croot and of Bloom. Using our…

We prove that any set $A\subset \mathbb{N}$ of positive upper density contains a finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$, answering a question of Erd\H{o}s and Graham.

We prove an old conjecture of Erd{\H o}s and Graham on sums of unit fractions: There exists a constant $b>0$ such that if we $r$-color the integers in $2,b^r]$, then there exists a monochromatic set $S$ such that $\sum_{n \in S} 1/n=1$.

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies…

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \cdots $ be any other positive integer sequence satisfying $ \sum_{i=1}^\infty \frac{1}{a_i} = 1 $. In this paper, we solve a…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

We count the number of subsets of $\{1,2,\cdots,n\}$ under different conditions and study the sequence obtained as we let $n$ increase.

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point…

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal…

We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the…

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this…

Propositions 1.1 -- 1.3 stated below contribute to results and certain problems considered in a paper by Erdos and Szekeres, on the behavior of products $\prod^n_1 (1-z^{a_j}), 1\leq a_1\leq \cdots\leq a_n$ integers. In the discussion,…

We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with…

For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A:…

Erd\"{o}s and Tur\'{a}n once conjectured that any set $A\subset\mathbb{N}$ with $\sum_{a\in A}{1}/{a}=\infty$ should contain infinitely many progressions of arbitrary length $k\geq3$. For the two-dimensional case Graham conjectured that if…

In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They…

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…