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1. On Single-Distance Graphs on the Rational Points in Euclidean Spaces

   https://doi.org/10.4153/S0008439520000181  

   Bau, Sheng; Johnson, Peter; Noble, Matt (March 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract For positive integers n and d > 0, let $$G(\mathbb {Q}^n,\; d)$$ denote the graph whose vertices are the set of rational points $$\mathbb {Q}^n$$ , with $$u,v \in \mathbb {Q}^n$$ being adjacent if and only if the Euclidean distance between u and v is equal to d . Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $$\mathbb {Q}^n$$ . In this paper, we show that a space $$\mathbb {Q}^n$$ has the property that all pairs of non-trivial distance graphs $$G(\mathbb {Q}^n,\; d_1)$$ and $$G(\mathbb {Q}^n,\; d_2)$$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $$G(\mathbb {Q}^n,\; d)$$ . 

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2. On the Strict Majorant Property in Arbitrary Dimensions

   https://doi.org/10.1093/qmath/haac021  

   Gressman, P T; Guo, S; Pierce, L B; Roos, J; Yung, P -L (July 2022, The Quarterly Journal of Mathematics)

   Abstract In this work we study d-dimensional majorant properties. We prove that a set of frequencies in $$\mathbb{Z}^d$$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all p > 0 if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least d + 2 frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $$p \not\in 2\mathbb{N}$$ of length 2. Any infinite set of frequencies in $$\mathbb{Z}^d$$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $$p \not\in 2\mathbb{N}$$ of length 2. Finally, given any p > 0 with $$p \not\in 2\mathbb{N}$$, we exhibit a set of d + 2 frequencies on the moment curve in $$\mathbb{R}^d$$ that violate the strict majorant property on $L^p([0,1]^d).$ 

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3. Zarankiewicz’s problem for semilinear hypergraphs

   https://doi.org/10.1017/fms.2021.52  

   Basit, Abdul; Chernikov, Artem; Starchenko, Sergei; Tao, Terence; Tran, Chieu-Minh (January 2021, Forum of Mathematics, Sigma)

   Abstract A bipartite graph $$H = \left (V_1, V_2; E \right )$$ with $$\lvert V_1\rvert + \lvert V_2\rvert = n$$ is semilinear if $$V_i \subseteq \mathbb {R}^{d_i}$$ for some $$d_i$$ and the edge relation E consists of the pairs of points $$(x_1, x_2) \in V_1 \times V_2$$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $$d_1 + d_2$$ variables for some s . We show that for a fixed k , the number of edges in a $$K_{k,k}$$ -free semilinear H is almost linear in n , namely $$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ for any $$\varepsilon> 0$$ ; and more generally, $$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$$ for a $$K_{k, \dotsc ,k}$$ -free semilinear r -partite r -uniform hypergraph. As an application, we obtain the following incidence bound: given $$n_1$$ points and $$n_2$$ open boxes with axis-parallel sides in $$\mathbb {R}^d$$ such that their incidence graph is $$K_{k,k}$$ -free, there can be at most $$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o -minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner). 

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4. A Duality Principle for Groups II: Multi-frames Meet Super-Frames

   https://doi.org/10.1007/s00041-020-09792-0  

   Balan, R.; Dutkay, D.; Han, D.; Larson, D.; Luef, F. (December 2020, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

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5. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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