arXiv subjects

Joshua Iosevich

Publications and source records attributed to Joshua Iosevich.

5 recordsLinked to original sources

arXiv2026

Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements

We construct an explicit 64-point subset of $\mathbb F_2^9$ which tiles by translation but has no linear-subspace tiling complement. Together with our earlier dimension-eight theorem and a product argument, this shows that every nonempty tile in $\mathbb F_2^d$ has a linear-subspace complement precisely when $d\le8$. The example is spectral, so it is not a counterexample to Fuglede's conjecture. Its construction uses six three-dimensional subspaces of $\mathbb F_2^6$, specified by a small table. We classify all its tiling complements and spectra: there are exactly 512 complements and 64 spectra, with one translation class of each. Every spectrum has normalized Wiener norm $15/8$. An appendix proves that Goyal's related construction is affinely inequivalent but has the same tiling complements and spectra. The construction and both classifications are proved from first principles. We also give elementary proofs of the positive assertion in dimensions at most seven and for all tile sizes other than sixteen in dimension eight. The remaining case uses our previously established theorem based on the classification of perfect binary codes of length fifteen.

math.CO↗

arXiv2026

Fuglede's conjecture holds in eight dimensions over the field with two elements

Fuglede's conjecture in a finite abelian group says that a nonempty subset tiles by translation if and only if it is spectral, meaning that the space of complex-valued functions on the subset has an orthogonal basis consisting of restrictions of group characters. We prove the conjecture in the eight-dimensional vector space over the field with two elements. In fact, every spectral set and every tile in this space has a tiling complement which is a linear subspace. We first prove that the Fourier support of a nonempty proper subset of density different from one half contains a two-dimensional subspace. Consequently, every spectral set of density one quarter has a subspace tiling complement. This gives a simpler proof of Kadowaki's dimension-seven result, replacing his use of the classification of Hadamard matrices of order 32 by the classical five classes of order 16. Our dimension-eight proof uses exact finite verifications involving Hadamard matrices of order 32 and perfect codes of length 15. We include the proofs of the reductions used from Kadowaki's preprint, so that neither positive result depends on his dimension-seven theorem. Finally, we give an elementary spectral example of density one eighth in dimension nine whose Fourier support contains no three-dimensional subspace. Thus, the Fourier-support argument at quarter density does not extend in this form to eighth density. Dimension nine is the only case still unresolved over the field with two elements, and this obstruction shows why the present Fourier-support method cannot resolve it as it stands.

math.CO↗

arXiv2026

Fuglede's conjecture holds in $\mathbb{F}_{p}^{d}$ for $p=2,\,d=8$ and $p=11,\,d=3$

Fuglede's conjecture in $\mathbb{F}_p^d$ says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in $\mathbb{F}_2^8$ and $\mathbb{F}_{11}^3$. Over $\mathbb{F}_2$ we show more generally that the Fourier support of a nonconstant indicator of density different from one half contains a plane. Consequently every spectral set of density one quarter has a subspace tiling complement. This supplies one of the dimension-eight reductions left by Kadowaki; the others are completed using classified Hadamard matrices and perfect binary codes. In $\mathbb{F}_2^9$ we prove that every tile is spectral and that every spectral set has power-of-two cardinality. All spectral cardinalities except $64$ are shown to tile, and we give further reductions of the remaining case. We also construct a spectral set of density one eighth whose Fourier support contains no three-dimensional subspace. For $\mathbb{F}_{11}^3$ we extend the geometric method of Fallon, Mayeli, and Villano by using moment identities, polynomial plane profiles, weighted incidence counts, and exact finite computations. The computations and their verification programs are described in the appendices.

math.CO↗

arXiv2026

Additive energy and Fourier complexity of graphs

We study the additive energy and the Fourier approximation of graph adjacency matrices labeled by a cyclic group. For a graph with $N$ vertices, $s$ edges and adjacency matrix $M$, the additive energy of its directed edge set is at most $\min\{(2s)^3,N^2\operatorname{tr}(M^4)\}$, and equality in the spectral bound holds exactly when the labeling attains the graph-energy lower bound for the Fourier ratio. Such a labeling also minimizes every best $k$-term Fourier approximation error, whose square equals the tail of the squared adjacency singular values. We prove that graphs of odd prime-power order admit an equality-attaining labeling exactly when they are cyclic Cayley graphs, and a connected example on $15$ vertices shows that equality need not imply vertex transitivity at other orders. We construct regular graph families with asymptotically maximal normalized additive edge energy and unbounded minimum Fourier ratio, one of them with density tending to one half. For dense graph sequences under uniformly random labelings, the collision dimension differs from the reciprocal squared edge density by a quantity tending to zero in probability. We also prove a quantitative stability estimate for the graph-energy bound and show why its dependence on a spectral scale cannot be omitted, even for regular graphs. For a rank-$m$ orthogonal projector we prove $m\leq D_4\leq\operatorname{FR}^2$, characterize the equality $D_4=m$ by cyclic translation invariance, and obtain a quantitative approximation by a circulant projector.

math.CO↗

arXiv2026

Edge complexity of graphs

Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.

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