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Paolo Leonetti

Publications and source records attributed to Paolo Leonetti.

At least 19 recordsLinked to original sources
arXiv2026

Nonpathological submeasures and $F_σ$ ideals

We study the relationship between nonpathological lower semicontinuous submeasures, $F_σ$ ideals on $ω$, and matrix summability ideals. First, we construct in $\mathsf{ZFC}$ a non-analytic ideal on $ω$ which is representable as an intersection of matrix summability ideals. Then, we show that an $F_σ$ ideal is nonpathological if and only if it can be represented as an intersection of summable ideals if and only if it can be represented as an intersection of matrix summability ideals. We also prove that the random graph ideal is pathological. Finally, we compute the exact degree of pathology of a certain submeasure related to a construction of Topsøe.

math.GN↗
arXiv2026

Tauberian rates and the Baire property of ideals

We study Tauberian conditions on the successive differences of real sequences which ensure that: (i) ordinary convergence coincides with ideal convergence; or (ii) every ordinary accumulation point is an ideal cluster point; or (iii) every ordinary accumulation point is an ideal limit point. First, we prove that an ideal $\mathcal{I}$ has the Baire property if and only if there exists a positive sequence $(r_n)$ tending to zero, with $\sum_n r_n=\infty$, such that $$ |x_{n+1}-x_n|=O(r_n) $$ is a Tauberian condition for any of the three previous statements. Second, for analytic $P$-ideals, we characterize the validity of the Tauberian rate $O(1/n)$ in terms of properties of lower semicontinuous submeasures which represent it. Lastly, in the case of statistical limit points, we prove that $o(1/n)$ is a Tauberian condition for the statement (iii), whereas $O(1/n)$ need not suffice: in this sense, $o(1/n)$ is an optimal Tauberian rate for (iii). On the other hand, for statistical cluster points in (ii), $O(1/n)$ is an optimal one among nonincreasing rates. For every positive null sequence $(r_n)$ with divergent sum, we also construct a summable ideal $\mathcal{I}$ for which $o(r_n)$ is an optimal Tauberian rate for both statements (ii) and (iii).

math.CA↗
arXiv2026

On the inverse Goldbach problem

We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq\mathbb N$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$ \frac{\sqrt{x}}{\log x}\ll A(x),B(x) \ll \sqrt{x}. $$ Second, we show that, if $A,B,C\subseteq\mathbb N$ have at least two elements each, then $$ ((A+B+C)\mathbin{\bigtriangleup}\mathbb P)(x) \gg \frac{\sqrt{x}}{(\log x)^2}. $$ An analogous estimate is obtained for sumsets of $k\ge 3$ sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.

math.NT↗
arXiv2026

Subadditive bijections on the positive reals

We show that there exists a subadditive bijection $f:(0,\infty)\to(0,\infty)$ such that $\liminf_{x\to 0}f(x)=0$ and $\limsup_{x\to 0}f(x)=1$. This answers a question by Matkowski and Swiatkowski in [Proc. Amer. Math. Soc. 119 (1993), 187--197].

math.CA↗
arXiv2026

On the density of $A+B$ when $B$ has few elements

Let $\mathsf{d}$ be the asymptotic density on the positive integers $\mathbb{N}^+$. We provide sufficient conditions on an infinite set $B\subseteq \mathbb{N}^+$ so that for each $α\in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=0$ and $\mathsf{d}(A+B)=α$. For instance, if $(b_n: n \in \mathbb{N}^+)$ is the increasing enumeration of $B$, it is sufficient that $\liminf_n b_{n+1}/b_n>1$. Moreover, we show that, if $|B\cap [1,n]|\ll \log(n)^{1-\varepsilon}$ for some $\varepsilon>0$, then for each $α\in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=\mathsf{d}(A+B)=α$. Conversely, we show that, for each $\varepsilon>0$, there exists an infinite set $B\subseteq \mathbb{N}^+$ such that $|B\cap [1,n]| \ll n^{2/3+\varepsilon}$ and $\mathsf{d}(A+B) \in \{0,1\}$ for every $A\subseteq \mathbb{N}^+$ such that $A+B$ admits asymptotic density.

math.NT↗
arXiv2026

Complexity and Polishability of characterized subgroups on the unit circle

Given an ideal $\mathcal I$ on $ω$, a subgroup $H$ of the unit circle $\mathbb T$ is said to be $\mathcal I$-characterized if there exists a sequence of integers $(a_n:n\inω)$ such that $$ H= \left\{ x\in\mathbb T: \mathcal I\text{-}\lim_{n\to\infty}a_nx=0 \right\}. $$ We investigate the descriptive complexity and Polishability of these subgroups in terms of the structural and topological properties of the ideal $\mathcal I$. Our main structural result shows that $\mathcal{I}$ is an analytic $P$-ideal if and only if all $\mathcal I$-characterized subgroups are Polishable. In such case, we explicitly describe a compatible finer Polish group topology. Using results on Polishable subgroups, we obtain a trichotomy for their possible Borel complexities. If $\mathcal{I}$ is a generalized density ideal, we show the sharper dichotomy that every proper $\mathcal I$-characterized subgroup is either countable or $F_{σδ}$-complete. We also prove that this fails for general analytic $P$-ideals by constructing a subgroup characterized by a summable ideal which is neither $F_σ$ nor $F_{σδ}$-complete. Finally, we give explicit descriptions of the subgroups associated with the sequences of powers, the Fibonacci sequence, and the sequence of factorials. We conclude with several open questions.

math.GN↗
arXiv2026

In orbit background for hard X-ray CubeSat polarimeters: case study of the CUSP mission in low-earth orbit

The CUbesat Solar Polarimeter (CUSP) project aims to measure the linear polarization of solar flares in the 25 - 100 keV hard X-ray band using a Compton scattering polarimeter. CUSP is a project in the framework of the Alcor Program of the Italian Space Agency aimed to develop innovative CubeSat technologies and missions. As part of CUSPs Phase B study, initiated in December 2024 and closed on July 2nd, 2026, estimating the in orbit background to optimize the signal-to-background ratio was one of the key objectives. In low-Earth orbit, the instrument is exposed to cosmic and albedo X-ray backgrounds, charged particles, and secondary radiation from the spacecraft and atmosphere. Simulating these contributions enables optimization of detector geometry and shielding to maximize signal-to-noise performance. We present initial in orbit background estimates for CUSP using a Geant4-based simulator. A detailed mass model of the CUSP has been implemented to simulate background components and estimate the background count rate in the CUSP orbit.

astro-ph.IM↗
arXiv2026

On non-symmetric $t$-convexity

Zsolt Páles asked whether there exists a function $f: \mathbb{R}\to \mathbb{R}$ such that $$ f\left(\frac{x+2y}{3}\right) \le \frac{f(x)+2f(y)}{3} \quad \text{ for all }x\le y $$ which is not midconvex. We show that the answer is negative.

math.GM↗
arXiv2026

Compton Polarimeter Prototype for the CUbesat Solar Polarimeter (CUSP) mission

The space-based CUbesat Solar Polarimeter (CUSP) mission aims to measure the linear polarization of solar flares in the hard X-ray band (25-100 keV) by means of a dual-phase Compton polarimeter. CUSP will allow to study the magnetic reconnection and particle acceleration in the flaring magnetic structures of our star with its unprecedented sensitivity to solar flare polarization. CUSP is a project under development as part of the Alcor Program of the Italian Space Agency aimed at developing new CubeSat missions. In the frame of CUSP's Phase B, which started in December 2024, a flight-representative prototype of the Compton polarimeter has been developed and characterized with hard X-ray sources in the laboratory. This prototype consists of a 4$\times$4 central matrix of plastic scintillator bars surrounded by 4 strips of 8 elongated GAGG scintillators, respectively coupled to a multi-anode photomultiplier tube and arrays of avalanche photodiodes. These sensors are read out by custom front-end electronics based on MAROC-3A and SKIROC-2A ASICs with a Xilinx Artix 7 FPGA. The plastic scintillators act as scatterers, while the GAGG bars fully absorb the scattered photons. Coincident plastic-GAGG events allow for reconstructing the Compton scattering direction, whose distribution allows for inferring the polarization parameters of the source. We report here the measured performance of the polarimeter prototype using well-known radioactive isotopes and X-ray tubes, allowing us to assess the performance of our polarimeter prototype over the full 25-100 keV energy range.

astro-ph.IM↗
arXiv2026

Linear relations modulo meager sets

We prove several topological zero-one laws. First, we show that, if a subset $A$ of a Banach space $X$ has the Baire property, then $A$ admits a somewhere dense set of vectors $x \in X$ such that $A+x$ agrees with $A$ modulo meager sets if and only if it is either meager or comeager. Additional equivalent conditions are given if $X=\mathbb{R}$. Second, we prove that if $A_1,\ldots,A_k\subseteq \mathbb{R}$ are subsets with the Baire property, $α_1,\ldots,α_k$ are nonzero reals with distinct finite sums, and $(x_{n,i}: n\in ω)$ are injective real sequences which converge to $0$ for each $i=1,\ldots,k$, then $$ \sum_{i=1}^k α_i(1_{A_i+x_{n,i}}-1_{A_i})=0 $$ modulo meager sets for all $n \in ω$ if and only if each $A_i$ is either meager or comeager. Finally, we provide some additional results in the case where the $α_1,\ldots,α_k$ do not have distinct finite sums. For instance, if $k=2$ and $α_1=α_2$, then the above equality holds modulo meager sets for all $n \in ω$ if and only if each $A_i$ is either meager or comeager or if $\{A_1,A_2\}$ is a partition of $\mathbb{R}$ modulo meager sets. Additional characterizations are given in the case $k\ge 3$. These results yield the category analogues of several results by Fejzić, Freiling, and Rinne in [J. London Math. Soc.~\textbf{82} (2010), no. 3, 717--732].

math.GN↗
arXiv2026

On a question of Ruzsa about densities of sumsets

Let $\mathsf{d}$, $\mathsf{d}^{\ast}$, and $\mathsf{d}_{\ast}$ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on $\mathbb N$. We show that there are no constants $0<ν<1$ and $c>0$ such that $$ \mathsf{d}_{\ast}(A+A)\ge c\bigl(\mathsf{d}^{\ast}(A+A)\bigr)^{1-ν}\mathsf{d}(A)^ν $$ for every set $A\subseteq \mathbb{N}$ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.

math.NT↗
arXiv2026

A characterization of Sophie Germain primes - II

Let $n\ge 4$ be an even integer. We show that $\{1^{σ(1)},2^{σ(2)},\ldots,n^{σ(n)}\}$ is a complete residue system modulo $n$ for some permutation $σ$ of $\{1,2,\ldots,n\}$ if and only if $p:=n/2$ is prime and $p-1$ is squarefree. This complements the results in [Int. J. Number Theory \textbf{14} (2018), 653--660], where an analogue characterization has been proved for odd integers.

math.NT↗
arXiv2026

Characterized subgroups on the unit circle

Given an ideal $\mathcal{I}$ on $ω$, a subgroup $H$ of the unit circle $\mathbb{T}$ is said to be $\mathcal{I}$-characterized if there exists an integer sequence $a=(a_n: n \in ω)$ such that $$ H=\mathsf{H}_{a}(\mathcal{I}):= \left\{x\in\mathbb{T}:\mathcal I\text{-}\lim_{n\to \infty} a_nx=0\right\}. $$ We also consider the corresponding $\mathcal{I}^\star$-version. We provide upper bounds for the topological complexities of those subgroups in terms of the complexity of $\mathcal{I}$. Moreover, we prove that Rudin--Keisler and Rudin--Blass reductions between ideals induce inclusions between the corresponding families of characterized subgroups. As a consequence, every characterized subgroup, and in particular every countable subgroup of $\mathbb{T}$, is $\mathcal{I}$-characterized for every meager ideal $\mathcal{I}$. We also show that if the image of $(a_n: n \in ω)$ contains arbitrarily large intervals, then every subgroup of $\mathbb{T}$ can be written as $\mathsf{H}_{a}(\mathcal{J})$ for some ideal $\mathcal{J}=\mathcal{J}_{H,a}$. We analyze the descriptive complexity and $P$-properties of these ideals. Finally, we study when the equality $\mathsf H_{a}(\mathcal{I})=\mathbb{T}$ forces $\mathrm{supp}(a)\in\mathcal{I}$. We prove this for a class of ideals satisfying a Katetov-type condition involving $\mathcal{ED}$, including nowhere tall ideals as well as the ideals $\mathsf{nwd}$ and $\mathsf{null}$. We also obtain non-inclusion results between families of $\mathcal{I}$-characterized subgroups: for instance, we show that if the ideal $\mathcal{I}$ is tall and translation invariant then the subgroup $\mathsf{H}_{(2^n)}(\mathcal{I})$ cannot be characterized. We use our results to answer several open problems posed in the literature.

math.FA↗
arXiv2026

On maximal families of independent sets with respect to asymptotic density

We study families of subsets of $ω$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that $\mathsf{d}[\mathcal{A}]$ attains a prescribed set of values in $(0,1)$ with at most countably many exceptions. In addition, under $\mathrm{cov}(\mathcal{N})=\mathfrak{c}$, it is possible to construct such $\mathcal{A}$ with no exceptions. We also construct $2^{\mathfrak{c}}$ maximal $\mathsf{d}$-independent families with pairwise distinct generated density fields and obtain maximal families with strong definability pathologies, including examples without the Baire property and, consistently, nonmeasurable examples.

math.LO↗
arXiv2026

Completeness and additive property for submeasures

Given an extended real-valued submeasure $ν$ defined on a field of subsets $Σ$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_ν$ defined by $d_ν(A,B):=\min\{1,ν(A\bigtriangleup B)\}$ for all $A,B \in Σ$ is complete. As an application, we show that if $φ: \mathcal{P}(ω)\to [0,\infty]$ is a lower semicontinuous submeasure and $ν(A):=\lim_n φ(A\setminus \{0, 1, \ldots, n-1\})$ for all $A\subseteq ω$, then $d_ν$ is complete. This includes the case of all weighted upper densities, fixing a gap in a proof by Just and Krawczyk in [Trans.~Amer.~Math.~Soc.~\textbf{285} (1984), 803--816]. In contrast, we prove that if $ν$ is the upper Banach density (or an upper density greater than or equal to the latter) then $d_ν$ is not complete. We conclude with several characterizations of completeness in terms of the Stone space of the Boolean algebra $Σ/ν$.

math.FA↗
arXiv2026

On rough ideal convergence

We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_η:η\in X\}$ of subsets of $X$ with $η\in F_η$, where each $F_η$ measures the allowed ``roughness'' of convergence toward $η$. More precisely, after introducing the corresponding notions of cluster and limit points, we prove several inclusion and invariance properties, discuss their structural properties, and give examples showing that the rough notions are genuinely different from the classical ideal ones.

math.GN↗
arXiv2026

Sets of Ramsey-limit points and IP-limit points

Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq ω$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in ω}$ taking values in $X$, let $Λ_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $Λ_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $Λ_{x}(FS)$ and of the type $Λ_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence.

math.GN↗
arXiv2026

Spectral performance of single-channel plastic and GAGG scintillator bars of the CUbesat Solar Polarimeter (CUSP)

Our Sun is the closest X-ray astrophysical source to Earth. As such, it makes a formidable case study to better understand astrophysical processes. Solar flares are in particular very interesting as they are linked to coronal mass ejections as well as magnetic field reconnection sites in the solar atmosphere. Flares can therefore provide insightful information on the physical processes at play on their production sites, but also on the emission and acceleration of energetic charged particles towards our planet, making it a formidable forecasting tool for space weather. While solar flares are critical to understanding magnetic reconnection and particle acceleration, their hard X-ray polarization -- key to distinguishing between competing theoretical models -- remains poorly constrained by existing observations. To address this, we present the CUbesat Solar Polarimeter (CUSP), a mission under development to perform solar flare polarimetry in the 25-100 keV energy range. CUSP consists of a 6U-XL platform hosting a dual-phase Compton polarimeter. The polarimeter is made of a central assembly of four 4x4 arrays of plastic scintillators, each coupled to multi-anode photomultiplier tubes, surrounded by four strips of eight elongated GAGG scintillator bars coupled to avalanche photodiodes. Both types of sensors from Hamamatsu are respectively read out by the MAROC-3A and SKIROC-2A ASICs from Weeroc. In this manuscript, we present the preliminary spectral performances of single plastic and GAGG channels measured in the laboratory using development boards of the ASICs foreseen for the flight model.

astro-ph.IM↗