Related papers: Generalizing the Clunie--Hayman construction in an…

We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{\pi_1}(n)\, A_{\pi_2}(n+h), $$ where $A_{\pi_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $\pi_i$ for…

Let $\tau_k$ be the $k$-fold divisor function. By constructing an approximant of $\tau_k$, denoted as $\tau_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and…

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $\delta\ge 1$ and usual height $H\ge 1$, and let $\alpha\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^\alpha$: for any…

We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_{\Gamma}B$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As…

The Erd\H{o}s-Rogers function $f_{s,t}$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. While good estimates for $f_{s,t}$ are known for some pairs $(s,t)$, notably when $t=s+1$, in…

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound…

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions $N\geq 5$. Then it was extended to lower dimensions $N\in \{3, 4\}$ by…

The Bohnenblust-Hille inequality for $m$-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as $m$ grows) is unknown, but significant for…

In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary…

In 1909, Hardy gave an example of a transcendental entire function, $f$, with the property that the set of points where $f$ achieves its maximum modulus, $\mathcal{M}(f)$, has infinitely many discontinuities. This is one of only two known…

We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erd\H{o}s and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min…

We investigate a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger…

Let $(R,\mathfrak m)$ be a local ring of characteristic $p>0$ and $M$ a finitely generated $R$-module. In this note we consider the limit: $\lim_{n\to \infty} \frac{\ell(H^0_{\mathfrak m}(F^n(M)))}{p^{n\dim R}} $ where $F(-)$ is the…

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $\Delta$-function, we derive lower bounds for the…

Let $\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and }…

For every $n\geq 2$, Bourgain's constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb{R}^n$ on which harmonic measure is defined. Jones and Wolff…

Let as usual $Z(t) = \zeta(1/2+it)\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is…