Papers

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \cdot \sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller–Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.

Let $\lambda$ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer $k\geq 2$, we prove that \[ \lim_{\min\{M,N\}\to \infty}\frac{1}{MN}\sum_{m=1}^M\sum_{n=1}^N\lambda(m)\lambda(m+a_1n)\cdots\lambda(m+a_kn)=0. \] We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.

We derive an explicit zero-side formula for the imaginary part of a smooth truncation of the logarithmic derivative of the Riemann zeta-function. The resulting kernel is the odd companion of the Fejér kernel in Montgomery's pair-correlation identity. Motivated by the recent breakthrough of Anthropic's Claude \cite{Claude26}, we also give an expository account of how the real-part identity can be turned into a matrix problem, overcoming the clustering obstruction for zeros off the critical line. Finally, we discuss how the new imaginary-part identity may contain additional information about such clustering.

Given an integer $k \geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$ and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.

For $\kappa > 1$, we study the reciprocal LCM-power function \[ \sigma^{[\kappa]}_{-1}(n) := \sum_{d \mid n}\mu(d)\,\sigma_{-1}(n / d)^\kappa. \] We determine its average and extremal orders. A natural Robin-scale normalization is strictly increasing in $\kappa$ and tends to $\sigma_{-1}$, yielding an RH-equivalent criterion. Unconditionally, the corresponding Robin inequality holds eventually on every bounded parameter interval. Directed interval arithmetic proves the inequality for each $n > 5040$ and $1 < \kappa \le 100{,}000$.

Assuming that $\zeta(s)$ has no nontrivial zero with $\mathrm{Re}\, s > 7/8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $\Lambda \le 13963/100000 = 0.13963$ for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width $1/5$. We derive an approximation on $3 \le \mathrm{Im}\, z \le 6$ from its general Riemann-Siegel coefficient bounds. At height $61/20$, a fixed cutoff permits two differentiations, and comparison with $\zeta$ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon's analytic inputs nor Gomila's new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.

Let $\mathcal D$ be a product of Drinfeld upper half spaces and let $J$ be the corresponding product of projective linear groups acting on $\mathcal D$. We prove that if $Y$ is a reduced, quasi-compact separated strict $\mathbb C_p$-analytic space and $f: Y\to\mathcal D$ is analytic with $\dim Y<\dim\mathcal D$, then $J f(Y(\mathbb C_p))$ is closed and nowhere dense in $\mathcal D(\mathbb C_p)$. The quasi-compactness assumption is necessary even when the source is a strict rigid analytic curve, while a quasi-compact perfectoid source can still have an orbit closure with nonempty interior. For every finite extension $F/\mathbb Q_p$ and $d\geq3$, we construct a closed smooth irreducible non-quasi-compact curve in $\Omega_F^{d-1}$ with dense orbit under a fixed diagonal one-parameter subgroup. We also construct a $\mathbb Q_p$-defined analytic map from the open unit disc to $\mathbf P^2$ with dense $\operatorname{PGL}_3(\mathbb Q_p)$-orbit, and an analytic map from a one-dimensional affinoid perfectoid torus to $\mathbf P^2$ whose $\operatorname{PGL}_3(\mathbb Q_p)$-orbit closure has nonempty interior.

We combine the universal polynomial matrix identities of Larsen–Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of $\text{SL}_2(\mathcal{O}_{K,S})$ for every global function field $K$ and every nonempty finite set of places $S$. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over $\mathbb{Z}$ that works simultaneously for all these rings.

This paper studies formal Laurent series, with particular emphasis on their continued-fraction and Laurent-series expansions. Knowledge of both expansions is important, for example, in the study of Kronecker-type sequences in uniform distribution theory and in the search for potential counterexamples to variants of Littlewood's conjecture over fields of formal Laurent series. Our approach investigates Hankel matrices over a field and relates them to Laurent-series expansions over the same field. We describe such matrices via a block $\mathcal{L}\mathcal{B}\mathcal{U}$ decomposition, using recursive algorithms based on the continued-fraction coefficients of the corresponding Laurent series. The upper-triangular factor $\mathcal{U}$ has several remarkable properties. In particular, it is constructed columnwise from the so-called Ostrowski-type representations of powers of $X$. Applying $\mathcal{U}$ in Niederreiter's digital construction yields an Ostrowski-type van der Corput sequence that can be related to the corresponding Kronecker-type sequence. For ordinary Kronecker sequences, such a relation is known in only one case, that of the $(n\varphi)$-sequence. The original motivation for our study of Hankel matrices was to recast the search for counterexamples to variants of Littlewood's conjecture over fields of Laurent series as the search for specific Kronecker-type sequences. Interestingly, the method developed for the study of Kronecker-type sequences also allows a new perspective on the celebrated Sobol' construction. Our approach yields a short and elegant proof of its excellent distribution properties.

For a positive integer $d$, let $\Delta_d^L$ denote the supremum of the densities of lattice sphere packings in $\mathbb R^d$. We prove that, for every sufficiently large $d$, $\Delta_d^L \gg d^2\log\log d\,2^{-d}$. Abuya-Gargava-Zhao obtained the same bound along a sparse sequence of dimensions by combining Venkatesh's cyclotomic symmetries with Klartag's stochastic ellipsoid method. We introduce a multi-cyclotomic gluing construction that allows this argument to work in every sufficiently large dimension.

We prove that for any prime $p$ and tame level $N$, the weight map from the cuspidal Hilbert eigenvariety constructed by Andreatta-Iovita-Pilloni for a totally real field $F$ to the weight space is proper in the sense of Buzzard-Calegari, under a non-vanishing hypothesis on certain Sen polynomials of Galois representations at the primes above $p$. In particular, this yields the properness of the eigenvariety away from non-positive integral Sen weights. Our approach extends the $p$-adic Hodge-theoretic argument of Diao-Liu.

We prove asymptotic formulas for the number of $m$-full ideals of bounded norm in a given ideal class and for the number of $m$-full algebraic integers of bounded Weil height.

We investigate how much arithmetic information about a variety can be detected by sections of the geometrically abelian fundamental groups of sufficiently small open subvarieties. Over a \(p\)-adic field, such a section determines a rational point on the Albanese torsor, and, under a conjectural assumption, this point is represented by a \(0\)-cycle of degree \(1\) on the original variety if the variety has index \(1\). For curves, the latter statement holds unconditionally. Over finitely generated fields, such sections still force the relative Brauer group to vanish away from the characteristic. The key input is an identification of the Chern class obstruction attached to a section with the obstruction arising from local Tate duality. As a byproduct, we give a new proof of an open-subvariety strengthening of a theorem of Esnault and Wittenberg.

Using the circle method, we prove that every sufficiently large positive integer is a sum of two primes, $u$ Fibonacci numbers and $5-u$ Lucas numbers for each $u\in\{1,2,3,4\}$. All recurrence indices are positive, and terms may be repeated.

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $\kappa$. In this paper, via varying $b$, $\ell$, and $\kappa$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+7}{18\xi_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $\xi_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188\xi^3-2160\xi^2+171\xi+190=0$. Explicitly, with $\arccos\in[0,\pi]$, \[ \xi_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2\pi}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3\xi_{\mathrm{new}}-1)(12\xi_{\mathrm{new}}^2-40\xi_{\mathrm{new}}+15)} {9(2\xi_{\mathrm{new}}+1)(24\xi_{\mathrm{new}}^2-20\xi_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6\xi_{\mathrm{new}}}{9+18\xi_{\mathrm{new}}},\qquad \kappa_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+5}{18\xi_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over $p$-adic fields with good reduction. In this paper, by slightly generalizing the $p$-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of $\mathbb{Q}$-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.

Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $\Delta(\tau,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $\sigma$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimes\sigma)_G$ is induced from $\Delta(\tau,i)|\det|^s$ and a fixed conjugate of $\sigma$, outside a finite set of values of $q_F^{-s}$ depending on $\tau$ and $\sigma$. No genericity assumption on $\sigma$ is needed. For irreducible $\sigma$, this determines all irreducible quotients of $J_s$ of the form $\sigma^\vee\boxtimes\pi$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $\sigma$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $\tau$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.

For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $\theta(s;z)=\sum_{m,n\in\mathbb Z}e^{-\pi s\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-\pi \big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2\pi(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<\alpha_a<\alpha_b<\alpha_c<1$ such that \begin{equation}\aligned\nonumber \Minima_{z\in\mathbb H,\,(a,b)\in\mathbb R^2}\Big(\theta(1;z)+\alpha J(z;a,b)\Big)= \begin{cases} \;\big(e^{i{\pi}/{3}};1/3,1/3\big), &\hbox{if}\;\; \alpha\in[0,\alpha_a),\\ \;\big(e^{i{\pi}/{3}};1/3,1/3\big)\;\hbox{or}\;\big(e^{i{\varphi_{\alpha_a}}};1/2,1/2\big), &\hbox{if}\;\; \alpha=\alpha_a,\\ \;\big(e^{i{\varphi_\alpha}};1/2,1/2\big),\;\varphi_\alpha\in(\varphi_{\alpha_a},\frac{\pi}{2}),\;\; &\hbox{if}\;\; \alpha\in(\alpha_a,\alpha_b),\\ \;\big(i;1/2,1/2\big),\; &\hbox{if}\;\; \alpha\in[\alpha_b,\alpha_c],\\ \;\big(iy_\alpha;1/2,1/2\big),\; y_\alpha\in(1,\sqrt3],\;\;y_1=\sqrt3,\;\;&\hbox{if}\;\; \alpha\in(\alpha_c,1], \end{cases} \endaligned\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose–Einstein condensates.

In 2008, Jacobi and Madden proved that $a^4+b^4+c^4+d^4=e^4$ has infinitely many primitive solutions with the linear relation $e=a+b+c+d$. We extend their result to relations with rational cubes $e=a+k^3(b+c+d)$ and prove infinitude for $k=3$ and $10$ other $k\ne 1$. We compile $30$ starting solutions. In the spirit of Jacobi and Madden's original paper, everything here is simple algebra except one theorem of Mazur.

This is the third paper in a series where we study arithmetic applications of the higher elliptic Gamma functions. In the first two articles in this series, we defined geometric families of these functions and proved that they satisfy coboundary and cocycle relations under the action of special linear groups and associated congruence subgroups. The main purpose of the present paper is to present a construction of conjectural higher elliptic units above number fields with exactly one complex place as special values of higher elliptic Gamma functions, upgrading the construction carried out by Bergeron, Charollois and García for complex cubic fields. These higher elliptic units are obtained by evaluating a multiplicative $(n-2)$-cocycle built from higher elliptic Gamma functions against a $(n-2)$-cycle associated to some group of totally positive units of a given number field of degree $n$ with exactly one complex place. We conjecture that these higher elliptic units are algebraic units which belong to prescribed abelian extensions of the base field where they are evaluated and that they satisfy a Kronecker limit formula which relates the logarithm of their modulus to values of derivatives of partial zeta functions at $s = 0$ in the base field. We showcase our conjecture on various examples for number fields of degree 3, 4 and 5.

Let $p>3$ be a prime number. If $A$ and $B$ are two principally polarized superspecial abelian surfaces over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$, we prove that there exists a separable polarized isogeny between them with multiplier at most $p/\sqrt{2}$. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, $\mathrm{KLPT}^{2}$ provides a heuristic algorithm with a multiplier upper bound of order $p^{6+o(1)}$ for arbitrary pairs and $p^{3+o(1)}$ when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below $p$ with an unconditional deterministic guarantee.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

We prove the hyperplane winning property of weighted badly approximable vectors inside nondegenerate analytic manifolds and affine subspaces satisfying almost optimal Diophantine conditions. This, in particular, answers a conjecture of Badziahin–Velani for higher-dimensional manifolds, including affine cases that were understood only in $\mathbb{R}^2$.

We show that there exists $c > 0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.

Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the étale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequality $\dim_{\mathbb{Q}_p} H^1_f(G_T, U_n) < \dim_{\mathbb{Q}_p}H^1_f(G_p, U_n)$ of Bloch–Kato Selmer schemes, with their ratio having limit superior at most $1/2$. Here $U_n$ is the lower central series quotient, and $T$ a finite set containing $p$ and the primes of bad reduction. This answers a particular case of a conjecture by Kim. Following the Iwasawa-theoretic method of Coates–Kim, we reduce the problem to bounding certain character multiplicities at degree $n$. We then define a twisted character lattice $B$ and its $p$-adic completion $B_p.$ Our main idea is to regard normalized multiplicities as $\mu_n(\lbrace x \in B_p: f(x,n)=0\rbrace)$ for a $p$-adic analytic function $f$ coming from Iwasawa theory and a sequence of measures $(\mu_n)_n$ weakly converging to the Haar probability measure $\mu$ on $B_p,$ which is the $p$-adic analogue of the uniform measure. We prove convergence by considering characters $\eta$ of $B_p$ and comparing $\int_{B_p} \eta \ d\mu_n$ to traces of operators induced by symplectic operators $h_\eta$, followed by a character formula. A compactness argument for sequences of $p$-adic loci gives our main Iwasawa estimate. Poitou-Tate duality, the global Euler characteristic formula, and Hodge filtration estimates then give the dimension inequality.

Let $S_k$ be the space of cusp forms of weight $k$ for ${\rm SL}_2(\mathbb Z)$, with $\dim S_k\geq1$, and let $\chi_p=\left(\frac{\cdot}{p}\right)$ for odd primes $p$. For fixed $s$ with ${\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\leq X$ satisfying $L(h,\chi_p,s)=0$ tends to zero as $X\to\infty$, uniformly over nonzero $h\in S_k$. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square $L$-values of the normalized Hecke eigenforms when ${\rm Re}(s)>k$. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in $S_k$ for the kernels associated with twisted standard and symmetric square $L$-functions, with explicit basis criteria in specified right half-planes when $\dim S_k=2$ or $3$. We show that almost every tuple of $\dim S_k$ primes gives a basis of $S_k^{\ast}$ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of $S_k$ obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.

Let $p_i$ be the $i$th prime, and let $f(N)$ be the largest cardinality of a subset of \[ \left\{\prod_{i=1}^N p_i^{a_i}:0\le a_i<N,\ a_i\in\mathbb{Z}\right\} \] containing no two distinct legs of an integer right triangle. We prove \[ f(N) \ll N^N (\log\log N)^{-1/1296}. \]

We prove uniform height gaps for subvarieties of abelian varieties in arbitrary characteristic, extending the new gap principle of Gao-Ge-Kühne. Our proof goes through the theory of adelic curves and globally valued fields, and proves the Bogomolov conjecture for an arbitrary globally valued field. This gives a different way to obtain uniform Bogomolov-type results, differing from the approaches of Dimitrov-Gao-Habegger-Kühne and Yuan. First, we prove the Bogomolov conjecture over globally valued fields for divisors. We then reduce the case of general subvarieties to the case of divisors by an induction argument. Specializing to the case of global function fields, we obtain a new proof of the geometric Bogomolov conjecture following the strategy of Gubler and Yamaki.

We show that the wild locus of an étale morphism of smooth adic spaces over an affinoid field is either empty or contains a divisorial point. As a consequence, we establish the equivalence of curve tameness and valuation tameness without using resolution of singularities. Moreover, given a finite étale map of regular schemes $f \colon Y \to X$ that is curve tame over a base $S$, we construct a compactification $\bar{f} \colon \overline{Y} \to \overline{X}$ over $S$ that is numerically tame.

We introduce an asymmetric variant of a classic problem by Roth about colourings of integers and midpoints. The new problem cannot be solved with the standard approach by Erdös-Sárközy-Sós which uses symmetry. We describe a Fourier Analytic approach which gives asymptotically tight bounds. In the $\mathbb{F}_3^n$ setting, the density increment we show is efficient and together with the Freiman-Ruzsa Theorem provides an affine subspace of near optimal codimension within the distinguished set. This approach adopted to the setting of the integers from $1$ to $N$ via Bohr sets and Bogolyubov-Ruzsa Lemma gives a progression of length being a power of $N$ which only depends on the number of colours. We then use Chang's Lemma and Balog-Szemerédi-Gowers Theorem to further improve the dependence on the number of colours.