Related papers: An improved lower bound for finite additive 2-base…

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N…

Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of…

We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the…

For an input graph $G$, an additive spanner is a sparse subgraph $H$ whose shortest paths match those of $G$ up to small additive error. We prove two new lower bounds in the area of additive spanners: 1) We construct $n$-node graphs $G$ for…

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

We estimate weighted character sums with determinants $ad-bc $ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d $ varying over the interval $ [1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as…

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…

In this paper, we consider some additive properties of integers with restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$ be three…

Let $$ F(x, y) = \prod\limits_{k = 0}^{n - 1}(\delta_kx - \gamma_ky) $$ be a binary form of degree $n \geq 1$, with complex coefficients, written as a product of $n$ linear forms in $\mathbb C[x, y]$. Let $$ h_F = \prod\limits_{k = 0}^{n -…

We give an elementary proof that the sum of the digits of $2^n$ in base 10 is greater than $\log_4 n$. In particular, the limit of the sum of digits of $2^n$ is infinite.

This work proves a Burgess bound for short mixed character sums in $n$ dimensions. The non-principal multiplicative character of prime conductor $q$ may be evaluated at any "admissible" form, and the additive character may be evaluated at…

Answering affirmatively a 2007 problem of Chen, the first author proved that there is a unique representation basis $A$ of $\mathbb{Z}$ and a constant $c>0$ such that $$ A(-x,x)\ge c\sqrt{x} $$ for infinitely many positive integers $x$,…

Given a set of integers $A$ and an integer $k$, write $A+k\cdot A$ for the set $\{a+kb:a\in A,b\in A\}$. Hanson and Petridis showed that if $|A+A|\le K|A|$ then $|A+2\cdot A|\le K^{2.95}|A|$. At a presentation of this result, Petridis…

The powers of 3 generate half of the odd residues mod 2^k (k>2), and a sign change yields the other half. In other words: 3 is a semi-primitive root of 1 mod 2^k (k>2). Hence each k-bit residue is n = +/- 3^i.2^j mod 2^k, with unique…

Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let $B_n$ be the number of nonisomorphic…

In this work we establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of "admissible" forms. This…

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight…

A dominating set on an $n $-dimensional hypercube is equivalent to a binary covering code of length $n $ and covering radius 1. It is still an open problem to determine the domination number $\gamma(Q_n)$ for $ n\geq10$ and $…

An additive system for the nonnegative integers is a family (A_i)_{i\in I} of sets of nonnegative integers with 0 \in A_i for all i \in I such that every nonnegative integer can be written uniquely in the form \sum_{i\in I} a_i with a_i \in…

We study the number of $k$-element sets $A \subset \{1,\ldots,N\}$ with $|A + A| \leq K|A|$ for some (fixed) $K > 0$. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a…