Related papers: New Upper Bound for Sums of Dilates

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…

Let $n \ge 2$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) =…

The sum $\lambda_1 + \lambda_n$ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer $k\geq 3$, let $\gamma_k$ denote the…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

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…

The Linear Arboricity Conjecture asserts that the linear arboricity of a graph with maximum degree $\Delta$ is $\lceil (\Delta+1)/2 \rceil$. For a $2k$-regular graph $G$, this implies $la(G) = k+1$. In this note, we utilize a network flow…

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. The problem of bounding $\lambda_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov,…

A set of non-negative integers A is an additive 2-basis with range n, if its sumset A+A contains 0, 1, ..., n but not n+1. Explicit bases are known with arbitrarily large size |A|=k and $n/k^2 \ge 2/7 > 0.2857$. We present a more general…

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where…

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If…

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots,…

Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq G$ be a random set of size $|G|^\alpha$. We let $$ \mu(A) = \max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|. $$ The issue is to determine…

Let $k$ and $N$ be positive integers with $k\ge2$ even. In this paper we give general explicit upper-bounds in terms of $k$ and $N$ from which all the residual representations $\bar{\rho}_{f,\lambda}$ attached to non-CM newforms of weight…

The purpose of this paper is to give explicit methods for bounding the number of vertices of finite $k$-regular graphs with given second eigenvalue. Let $X$ be a finite $k$-regular graph and $\mu_1(X)$ the second largest eigenvalue of its…

Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its…

Let $A$ be a finite subset of an abelian group $G$, and suppose that $|A+A|\leq K|A|$. We show that for any $\epsilon>0$, there exists a constant $C_\epsilon$ such that $A$ can be covered by at most $\exp(C_\epsilon \log(2K)^{1+\epsilon})$…

Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain…

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use…

For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA =\{a_1+... a_l| a_i \in A_i \}. $$ A closely related and equally interesting notion is that of…

We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k}…