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1. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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2. Bell state analyzer for spectrally distinct photons

   https://doi.org/10.1364/OPTICA.443302  

   Lingaraju, Navin_B; Lu, Hsuan-Hao; Leaird, Daniel_E; Estrella, Steven; Lukens, Joseph_M; Weiner, Andrew_M (March 2022, Optica)

   We demonstrate a Bell state analyzer that operates directly on frequency mismatch. Based on electro-optic modulators and Fourier-transform pulse shapers, our quantum frequency processor design implements interleaved Hadamard gates in discrete frequency modes. Experimental tests on entangled-photon inputs reveal fidelities of $\sim <\#comment/>98\mathrm{\%<\#comment/>}$for discriminating between the $|{\mathrm{\Psi <\#comment/>}}^{+}⟩<\#comment/>$and $|{\mathrm{\Psi <\#comment/>}}^{-<\#comment/>}⟩<\#comment/>$frequency-bin Bell states. Our approach resolves the tension between wavelength-multiplexed state transport and high-fidelity Bell state measurements, which typically require spectral indistinguishability. 

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3. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo; Wang, Jinhuan (June 2024, Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

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4. Bohr recurrence and density of non-lacunary semigroups of ℕ

   https://doi.org/10.1090/proc/17006  

   Frantzikinakis, Nikos; Host, Bernard; Kra, Bryna (January 2025, Proceedings of the American Mathematical Society)

   A subset $R$of integers is a set of Bohr recurrence if every rotation on ${\mathbb{T}}^d$returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!2^m{3}^n:<\#comment/>k,m,n\in <\#comment/>\mathbb{N}\}$is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$is a real polynomial with at least one non-constant irrational coefficient, then the set $\left\{P\right(2^m{3}^n):<\#comment/>m,n\in <\#comment/>\mathbb{N}\}$is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl. 

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5. Spectroscopic study of the 4 f ⁷ 6 s ^{2 8} S 7/2∘−4 f ⁷ ( ⁸ S ^∘ )6 s 6 p ( ¹ P ^∘ ) ⁸ P _9/2 transition in neutral europium-151 and europium-153: absolute frequency and hyperfine structure

   https://doi.org/10.1364/JOSAB.467968  

   Herd, M_T; Maruko, C.; Herzog, M_M; Brand, A.; Cannon, G.; Duah, B.; Hollin, N.; Karani, T.; Wallace, A.; Whitmore, M.; et al (September 2022, Journal of the Optical Society of America B)

   We report on spectroscopic measurements on the $4f^{7}6s^2{}^8{S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4f^7{(}^8{S}^{\circ <\#comment/>}\left)6s6p{(}^1{P}^{\circ <\#comment/>}\right){}^8{P}_{9/2}$transition in neutral europium-151 and europium-153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the $6s6p{(}^1{P}^{\circ <\#comment/>}){}^8{P}_{9/2}$state were found to be $\mathrm{A}\left(151\right)=-<\#comment/>228.84\left(2\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(151\right)=226.9\left(5\right)\mathrm{M}\mathrm{H}\mathrm{z}$and $\mathrm{A}\left(153\right)=-<\#comment/>101.87\left(6\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(153\right)=575.4\left(1.5\right)\mathrm{M}\mathrm{H}\mathrm{z}$, which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results. 

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