arXiv subjects

Christina Giannitsi

Publications and source records attributed to Christina Giannitsi.

7 recordsLinked to original sources

arXiv2026

Sharp Thresholds for Distance Patterns in Random Subsets of $\mathbb Z^d$

Let $d \geq 5$, $0 < γ< d - 2$, and $Ω_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-γ}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq Ω_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \lesssim λ\lesssim N^2$, a squared distance of the form $q^2 λ$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.

math.CO↗

arXiv2026

Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$

We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along $Ω$ converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector--disk comparison in an explicit logarithmic shrinking range, with a power-saving error in $\log N$, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and $T^{Ω(n)} x$, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.

math.DS↗

arXiv2024

Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $ω\subset \mathbb{T}$. There is an integer $N_ω$, so that every odd integer $n$ with $N(n)>N_ω$ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus ω) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in ω$, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.

math.NT↗

arXiv2024

Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(T^n x) $$ converges for every $f\in L^p$, where $\mathscr{N} (n) = n \bar{n}$, and $D(N) = \sum _{\mathscr{N} (n) \leq N} d(n) $, and $1<p\leq \infty$. To do so we study the averages $$ A_N f (x) = \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n) \,f(x-n) ,$$ and obtain improving and weighted maximal inequalities for our operator, in the process.

math.CA↗

arXiv2023

Lower Estimate on Square Function of an Indicator Set

Let $Sf$ be a discrete martingale square function. Then, for any set $V$ of positive probability, we have $\mathbb{E} S(\mathbf{1}_V)^2 \geq η\mathbb{P}(V)$ for an absolute constant $η>0$. We extend this to wavelet square functions, and discuss some related open questions.

math.CA↗

arXiv2022

Improving and Maximal Inequalities for Primes in Progressions

Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along the primes in a progression of diameter $ y$, given by integer $ (b,y)=1 $. \begin{align*} A_{N,y,b} := \frac{ϕ(y)}{N} \sum _{\substack{n N _{y,r}} \lvert A_{N,y,b} f \rvert \rVert_{r}\ll \lVert f\rVert_{r}. \end{align*} The implied constant is only a function of $ r$. The uniformity over progressions imposes several novel elements on the proof.

math.CA↗

arXiv2022

Averaging with the Divisor Function: $\ell^p$-improving and Sparse Bounds

We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$ where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$, that is $$ \left( \frac{1}{N} \sum |K_Nf|^{p'} \right)^{1/p'} \lesssim \left(\frac{1}{N} \sum |f|^p \right)^{1/p} $$ as long as $f$ is supported on $[0,N]$. We also show that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse founds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

math.CA↗