July 15, 2026
The Beck–Fiala conjecture asserts that every matrix \(A\in\{0,1\}^{n\times T}\) with at most \(d\) nonzero entries in each column has discrepancy \(O(\sqrt d)\). A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for \(d \ge \log(T)^2\). The present article extends the validity of the classical offline Beck–Fiala conjecture to \(d \ge \log(T)^{1+o(1)}\); moreover, the main thrust of the result is that it is actually obtained by an efficient online algorithm that minimizes prefix discrepancy. The result is also essentially optimal, since online prefix discrepancy is known to scale as \(\omega(\sqrt{d})\) for \(d =o(\log T)\). As an immediate corollary, the open question of online vector balancing in the Spencer setting is also resolved.
The algorithm is based on a compactly supported Metropolis fixed-point walk, constructed by combining ideas from several recent works on the online Komlós problem. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.
The offline Beck–Fiala conjecture asserts that every binary matrix \(A \in \{0,1\}^{n \times T}\) with \(d\)–sparse columns has discrepancy \(\mathrm{disc}(A) := \min_{\varepsilon\in \{-1,1\}^T} \|A\varepsilon\|_\infty\) of order \(O(\sqrt{d})\). Crucially, the conjectured bound does not depend on the dimensions \(n\) and \(T\). The classical Beck–Fiala theorem [1] asserts an upper-bound of order \(d\), while a celebrated result of Banaszczyk [2] gives the bound \(O(\sqrt{d\log T})\), which provides an improvement in the superlogarithmic regime \(d \gg \log T\). Recent breakthroughs of Bansal and Jiang [3] establish the conjectured \(O(\sqrt d)\) bound for \(d \ge \log(T)^2\); additionally, for any sparsity, they show the uniform bound of \[\widetilde{O}(\sqrt d+\sqrt{\log T})\,,\] where polynomial factors of \(\log\log T\) are suppressed in the \(\widetilde{O}\) notation. This remains the current state of the art.
In the closely related problem of algorithmic online discrepancy minimization, a sequence of hidden vectors \(a_1,\dots,a_T\) are fixed ahead of time by an oblivious adversary. Then, the vectors are sequentially revealed to the algorithm, which must then immediately and irrevocably chose signs \(\varepsilon_1,\dots, \varepsilon_T\) in order to minimize \(\|\sum_{i \le T} \varepsilon_i a_i\|_\infty\). Here, “online” refers to the algorithmic restriction that \(\varepsilon_t\) is chosen just as a function of \(a_1,\dots,a_t\), while “oblivious” refers to the fact that the vectors \(a_1,\dots,a_T\), although hidden from the algorithm, are fixed ahead of time and hence cannot be adversarially adapted to the algorithms choices. Finally, an algorithm which further guarantees bounds on \[\max_{t \le T} \left\|\sum_{1 \le s \le t} a_s \varepsilon_s\right\|_\infty\] is said to minimize “prefix” discrepancy, as opposed to “terminal” discrepancy.
Three canonical regimes provide the main focus of the combinatorial discrepancy literature: the Spencer setting has each vector \(a_i\) satisfying \(a_i \in [-1,1]^n\); the Beck–Fiala setting has \(a_i \in\{0,1\}^n\) and \(\|a_i\|_0 \le d\) for a parameter \(d := d(n)\); the Komlós setting has each vector \(a_i \in \mathbb{R}^n\) and \(\|a_i\|_2 \le 1\). Roughly, the state of the art for online vector balancing is a result of Kulkarni, Reis, and Rothvoss [4], which proves the optimal general Komlós bound \(O(\sqrt{\log T})\). This bound was very recently made algorithmic by Aden-Ali [5]. Applied directly to the Beck–Fiala setting, this result yields an online prefix discrepancy upper bound of \[\label{eq:direct-online-komlos} O\!\left(\sqrt{d \log T}\right)\,.\tag{1}\] This is, to our knowledge, the best available bound for the online Beck–Fiala problem.
The main contribution of this article is an efficient online algorithm that achieves the optimal bound of \(O(\sqrt{d})\) prefix discrepancy—which incidentally establishes the offline Beck–Fiala conjecture—down to nearly logarithmic values of the sparsity parameter \(d\).
Theorem 1 (Online Beck–Fiala). For every fixed \(\eta>0\) there are constants \(c_\eta,C_\eta>0\) with the following property. Let \(2\leq d\leq n\). There is a randomized online algorithm such that for any \(T\) and any fixed sequence \(a_1,\ldots,a_T\in[-1,1]^n\) in which every column has at most \(d\) nonzero entries, \[\label{eq:main-tail} \mathbb{P}\left\{ \max_{t\leq T}\Big\|\sum_{s\le t}\varepsilon_sa_s\Big\|_\infty >C_\eta\sqrt d \right\} \leq C_\eta\, T \exp\left(-\frac{c_\eta\, d}{\log^{2+\eta}(e\,d)}\right).\tag{2}\] Consequently, there are constants \(C,C',T_0 > 0\) depending only on \(\eta\) such that the prefix discrepancy of any such \(a_1,\dots,a_T\) is at most \(C'\sqrt{d}\) whenever \(T \ge T_0\) and \[\label{eq:main-threshold} d\geq C (\log T) (\log\log T)^{2+\eta}.\tag{3}\]
We collect two immediate consequences of this result in relation to existing conjectures.
Corollary 1 (Offline Beck–Fiala conjecture). For any matrix \(A \in \{0,1\}^{n \times T}\) with each column having support size of at most \(d\), where \(d\) satisfies 3 , it holds: \[\mathrm{disc}(A) = O_\eta\big(\sqrt{d}\big)\,.\]
Next, it follows that online prefix discrepancy in the Spencer setting is bounded by \(\sqrt{n}\). This answers Conjecture 1 of [4] affirmatively and in strong form. Previously, such bounds were only known in the “average case” setting where the incoming vectors are sampled uniformly at random from \(\{-1,+1\}^n\) [6].
Corollary 2 (Online prefix Spencer). For every \(\eta>0\) there are constants \(c_\eta,C_\eta>0\) and \(n_0(\eta)\) such that the following holds. If \(n\geq n_0(\eta)\) and \[T\leq \exp\left(\frac{c_\eta n}{\log^{2+\eta}(en)}\right),\] then there is a randomized online algorithm such that, for every fixed sequence \(a_1,\ldots,a_T\in[-1,1]^n\), \[\mathbb{P}\left\{ \max_{t\leq T} \left\|\sum_{s=1}^t\varepsilon_sa_s\right\|_\infty \leq C_\eta\sqrt n \right\} \geq 1-\exp\left[-\frac{c_\eta n}{\log^{2+\eta}(en)}\right].\] In particular, taking \(T=n\) gives a positive answer to Conjecture 1 of [4], with the stronger guarantee holding simultaneously for every prefix.
Finally, we observe that the range of \(d\) considered in 3 is essentially optimal from the perspective of online discrepancy minimization.
Remark 2 (Kulkarni-Reis-Rothvoss [4], Theorem 4). There is a strategy for an oblivious adversary that produces a sequence of vectors \(a_1,\dots,a_T \in [-1,1]^2\) such that the prefix discrepancy incurred by any online algorithm is, with probability at least \(1-2^{-T^{\Omega(1)}}\), at least \(\Omega(\sqrt{\log T})\). In particular, no online algorithm can guarantee \(O(\sqrt d)\) prefix discrepancy throughout the sublogarithmic regime \(d=o(\log T)\), even against an oblivious adversary.
The interested reader is referred to the survey [7] for general background on the rich field of discrepancy theory, both online and offline. We now introduce some specific algorithmic developments in the study of the online Komlós problem that are closely related to the present article.
Alweiss, Liu, and Sawhney introduced the self-balancing walk, obtaining \(O(\!\sqrt{\log(nT)})\)-sub-Gaussian prefix sums and an \(O(\log(nT))\) online Komlós discrepancy bound in linear time [8]. Liu, Sah, and Sawhney then constructed a Gaussian fixed-point walk whose increments lie in \(\{-1,0,1\}\), as well as a variant with increments in \(\{-1,1,2\}\); these yield online Banaszczyk-type bounds for partial colorings and relaxed signings [9]. Kulkarni, Reis, and Rothvoss obtained genuine signs and constant-sub-Gaussian prefixes, giving the optimal \(O(\sqrt{\log(eT)})\) discrepancy bound, although with exponential running time [4].
Subsequently, a work of Smirnov and Vershynin [10] took a similar approach to [9] and construct a random walk that can be kept inside of a convex body by rejecting or discarding a controlled fraction of random steps. Aden-Ali showed that three suitably balanced Gaussian fixed-point walks can instead be coupled so that their increments sum to a genuine sign, recovering the optimal Kulkarni–Reis–Rothvoss bound in linear time [5]. Our proof combines this triplet-coupling idea with a compact invariant law tailored to sparse arrivals.
In view of the above works, the authors prompted ChatGPT 5.6 Pro to solve the online Spencer problem by applying the coupling of [5] to the bounded random walk of [10]. A complete proof for \(O(\sqrt n)\) online prefix discrepancy with \(T=n\) was produced in a single response. The choice of the bump density in Section 3 was generated by GPT rather than from the authors’ initial proof strategy. Subsequent rounds of prompting extended the proof to the online Beck–Fiala setting and to longer horizons. The authors manually verified and rewrote the proofs and organized the results into the present manuscript.
Let \(\rho\) be a positive probability density on \((-L,L)\), to be defined later, extended to be zero outside the interval. Define \[\mu:=\rho^{\otimes n}, \qquad f(x):=\prod_{i=1}^n\rho(x_i).\] Equivalently, \(f(x)= e^{-U(x)}\) on \((-L,L)^n\), where \[U(x)=-\sum_{i=1}^n \log\rho(x_i),\quad x\in (-L,L)^n,\] with the convention \(U(x)=+\infty\) outside the cube. For any \(x\in (-L,L)^n\) and arbitrary \(v\in\mathbb{R}^n\), set \[\begin{align} p_v^+(x)&:=\frac{1}{3}\min\left\{1,\frac{f(x+v)}{f(x)}\right\}, \tag{4}\\ p_v^-(x)&:=\frac{1}{3}\min\left\{1,\frac{f(x-v)}{f(x)}\right\}, \tag{5}\\ p_v^0(x)&:=1-p_v^+(x)-p_v^-(x). \tag{6} \end{align}\] Given a non-random sequence of vectors \(v_0,v_1,\dots\) in \(\mathbb{R}^n\), we consider \(n\)–dimensional random walk \(X_0,X_1,\dots,\) where \(X_0\sim\mu\) and \(X_{t}\) is defined through \[X_t=X_{t-1}+\delta_t\,v_t,\] with \(\delta_t\) sampled in \(\{-1,0,+1\}\) via \[\mathbb{P}\{\delta_t=s\;|\,X_{t-1}\big\}=p_{v_t}^s(X_{t-1}).\]
Remark 3. Note that the transition probabilities at step \(t\) are determined by the input vectors \(v_1,\dots,v_t\), i.e., are computable in an online fashion.
Lemma 1 (Stationarity). For every non-random sequence of vectors \(v_0,v_1,\dots\) in \(\mathbb{R}^n\), \(X_t\sim\mu\).
Proof. Assume \(X_{t-1}\sim \mu\). For every measurable subset \(S\) of \((-L,L)^n\), \[\begin{align} \mathbb{P}\{X_t\in S\} &=\int\limits_{(-L,L)^n} \big( p_{v_t}^+(x)\,{\boldsymbol{1}}_{\{x+v_t\in S\}} + p_{v_t}^-(x)\,{\boldsymbol{1}}_{\{x-v_t\in S\}} + p_{v_t}^0(x)\,{\boldsymbol{1}}_{\{x\in S\}} \big)\,f(x)\,dx\\ &= \int\limits_{(-L,L)^n} f(x)\,{\boldsymbol{1}}_{\{x\in S\}} \,dx, \end{align}\] implying the claim. ◻
Definition 1 (Balanced state). A vector \(x\in(-L,L)^n\) is \(v\)-balanced if \[\label{eq:balanced} p_v^+(x)+p_v^-(x)\geq\frac{1}{3}.\tag{7}\]
Definition 2 (Curvature). For \(x\) satisfying both \(x + v\in(-L,L)^n\) and \(x - v\in(-L,L)^n\), define \[\Delta_\pm(x,v):=U(x\pm v)-U(x), \qquad D_v(x):=\Delta_+(x,v)+\Delta_-(x,v).\]
We will utilize the following convenient sufficient condition for balance.
Lemma 2 (Curvature criterion). If \(x+v\in(-L,L)^n\), \(x-v\in(-L,L)^n\), and \(D_v(x)\leq2\log2\), then \(x\) is \(v\)-balanced.
Proof. We have \[D_v(x)=(U(x + v)-U(x))+(U(x - v)-U(x)).\] If \(U(x + v)-U(x) \leq 0\) then \(f(x+v)\geq f(x)\), and \(p_v^+(x)=1/3\). Similarly, if \(U(x - v)-U(x) \leq 0\) then \(f(x-v)\geq f(x)\), and \(p_v^-(x)=1/3\). Finally, if both increments \(U(x + v)-U(x)\) and \(U(x - v)-U(x)\) are positive then, in view of convexity of \(s\to \exp(-s)\), \[p_v^+(x)+p_v^-(x) =\frac{1}{3}\left(e^{-\Delta_+(x,v)}+e^{-\Delta_-(x,v)}\right) \geq\frac{2}{3}e^{-D_v(x)/2}\geq\frac{1}{3}.\] ◻
We use the following finite coupling lemma, in the form established by Aden-Ali [5].
Lemma 3 (Three-way coupling; [5]). For \(j\in\{1,2,3\}\), let \(a_j,b_j\) satisfy \[0\leq a_j,b_j\leq\frac{1}{3}, \qquad a_j+b_j\geq\frac{1}{3}.\] There is a coupling \((\delta_1,\delta_2,\delta_3)\) of random variables in \(\{-1,0,1\}\) such that \[\mathbb{P}\{\delta_j=1\}=a_j, \qquad \mathbb{P}\{\delta_j=-1\}=b_j,\] for every \(j\), and \[\delta_1+\delta_2+\delta_3\in\{-1,1\} \qquad\text{almost surely}.\]
For each choice of invariant density \(\rho\), we are able to construct a well-behaved random walk in an online fashion. We now extract the precise corresponding algorithmic result for arbitrary \(\rho\), which will later inform our actual choices for \(\rho\).
Proposition 4. Let \(\mathcal{V}\subseteq\mathbb{R}^n\) be a family of column vectors. Suppose a product law \(\mu\) supported on \((-L,L)^n\) satisfies \[\label{eq:generic-beta} \sup_{v\in\mathcal{V}} \mathbb{P}_{X\sim\mu}\{X\text{ is not v-balanced}\} \leq\theta.\qquad{(1)}\] Then there is a randomized online vector balancing algorithm such that for every fixed sequence \(v_1,\ldots,v_T\in\mathcal{V}\), \[\label{eq:generic-conclusion} \mathbb{P}\left\{ \max_{k\leq T}\left\lVert \sum_{t \le k}\varepsilon_tv_t\right\rVert_\infty>6L \right\} \leq3T\theta.\qquad{(2)}\]
Proof. We construct three auxiliary coupled random walks \[(X_{t,j})_t,\quad j=1,2,3,\] with each \((X_{t,j})_t\) defined as in Subsection 2.1; namely, \(X_{0,j}\sim \mu\), with transition probabilities computed via 4 , 5 , 6 . The coupling is produced as follows. The algorithm maintains a failure flag initially set to false. At step \(t\geq 1\), compute the six probabilities \(p_{v_t}^\pm(X_{t-1,j})\).
As long as the flag is unset and all three previous states \(X_{t-1,j}\) are \(v_t\)-balanced, apply Lemma 3 to sample the coupled variables \(\delta_{t,j}\) with the marginals \[\mathbb{P}\{\delta_{t,j}=s\;|\,X_{t-1,j}\big\}=p_{v_t}^s(X_{t-1,j}),\quad s=\pm,0,\] so that \[\varepsilon_t:=\delta_{t,1}+\delta_{t,2}+\delta_{t,3}\in\{-1,1\},\] and set \[X_{t,j}=X_{t-1,j}+\delta_{t,j}v_t,\quad j=1,2,3.\] If at least one of \(X_{t-1,j}\)’s is not \(v_t\)-balanced, set the failure flag, sample the three increments \(\delta_{t,j}\) independently according to their one-walk marginals, and output the corresponding \(X_{t,j}\)’s and an arbitrary legal sign \(\varepsilon_t\). After the flag has been set, continue to evolve all three states independently according to their one-walk transition probabilities.
Applying Lemma 1, we obtain that regardless the choice of non-random vectors \(v_t\), all \(X_{t,j}\)’s are distributed according to \(\mu\). A union bound using ?? shows that some state is unbalanced at some time \(t\leq T\) with probability at most \(3T\theta\). On the complementary event, the failure flag is never set and, for every prefix \(k\), \[\begin{align} \sum_{t=1}^k\varepsilon_tv_t =\sum_{j=1}^3\sum_{t=1}^k\delta_{t,j}v_t =\sum_{j=1}^3(X_{k,j}-X_{0,j}). \end{align}\] Every state remains in \((-L,L)^n\), so the last expression has infinity norm at most \(6L\), simultaneously for all \(k\). ◻
Remark 5 (Obliviousness). The proof uses the obliviousness of the adversary: \(v_t\) is fixed independently of the current private state of the algorithm. For an adaptive adversary, conditioning on earlier output signs can bias the law of \(X_{t-1,j}\), and ?? cannot be applied in this form.
Finally, we conclude with a key result that makes verification of the balance condition tractable.
Lemma 4 (Reduction to equal weights via majorization). Let \(d\geq2\) and \(M\geq1\) be integers, let \((Z_i)_{i\geq1}\) be independent copies of a nonnegative random variable \(Z\) with \(\mathbb{E}Z^2<\infty\), and let \(w_1,\ldots,w_M\) satisfy \[\label{eq:flat-weights} 0\leq w_i\leq\frac{1}{d}, \qquad \sum_{i=1}^Mw_i\leq1.\tag{8}\] Define \(S_w=\sum_{i=1}^Mw_iZ_i\) and \(\overline{Z}_d=d^{-1}\sum_{i=1}^dZ_i\). If \(K\geq0\) and \[\mathbb{P}\{\overline{Z}_d>K\}\leq\varepsilon,\] then \[\label{eq:majorization-tail} \mathbb{P}\{S_w>K+1\} \leq (\mathbb{E}\overline{Z}_d^2)^{1/2}\varepsilon^{1/2}.\tag{9}\]
Proof. Choose \(N\geq M\) and extend \((w_1,\ldots,w_M)\) to \(\widetilde{w}=(\widetilde{w}_1,\ldots,\widetilde{w}_N)\) so that \[\widetilde{w}_i=w_i\quad(1\leq i\leq M), \qquad 0\leq\widetilde{w}_i\leq\frac{1}{d}, \qquad \sum_{i=1}^N\widetilde{w}_i=1.\] This is done by appending weights of size \(1/d\), followed if necessary by one smaller weight. Necessarily \(N\geq d\). Since \(Z_i\geq0\), \[S_w\leq\widetilde{S}_w:=\sum_{i=1}^N\widetilde{w}_iZ_i.\]
Let \(p_i=d\widetilde{w}_i\), and let \(s_i=\sum_{j\leq i}p_j\) with \(s_0=0\). The intervals \(J_i=[s_{i-1},s_i)\) partition \([0,d)\) and have length at most one. If \(U\) is uniform on \([0,1)\) and independent of the \(Z_i\), let \[I(U)=\{i:J_i\cap\{U,U+1,\ldots,U+d-1\}\ne\varnothing\}.\] The displayed \(d\) points lie in distinct intervals, so \(|I(U)|=d\). Moreover, \(\mathbb{P}_U\{i\in I(U)\}=|J_i|=p_i\). Thus, conditionally on the \(Z_i\), \[\widetilde{S}_w =\mathbb{E}_U\left[\frac{1}{d}\sum_{i\in I(U)}Z_i\right].\] For the increasing convex function \(\phi(x)=(x-K)_+\), conditional Jensen and exchangeability give \[\mathbb{E}\phi(S_w)\leq\mathbb{E}\phi(\widetilde{S}_w) \leq\mathbb{E}\phi\left(\frac{1}{d}\sum_{i\in I(U)}Z_i\right) =\mathbb{E}\phi(\overline{Z}_d).\] On \(\{S_w>K+1\}\) the left-hand integrand is at least one. On the other hand, Cauchy–Schwarz gives \[\mathbb{E}(\overline{Z}_d-K)_+ \leq (\mathbb{E}\overline{Z}_d^2)^{1/2} \mathbb{P}\{\overline{Z}_d>K\}^{1/2}.\] Combining the inequalities proves 9 . ◻
We now construct the stationary measure. Fix \(\beta>0\). For \(-1<y<1\) define \[\label{eq:bump-potential-main} S(y):=1-y^2, \qquad A(y):=S(y)^{-\beta}, \qquad \Phi_\beta(y):=e^{A(y)},\tag{10}\] and let \(Y\) be a random variable with density \[\label{eq:bump-density-main} h_\beta(y):=Z_\beta^{-1}e^{-\Phi_\beta(y)}\mathbf{1}\{|y|<1\}.\tag{11}\] Further define \[\label{eq:bump-q-main} q=q_\beta:=2+\frac{2}{\beta}, \qquad \ell(x):=\log^q(ex),\quad x\geq1.\tag{12}\] The product law is obtained by scaling \(Y\) by a constant support radius \(L=L_\beta\) chosen below.
Direct differentiation gives for every \(y\in(-1,1)\): \[\label{eq:bump-second-main} 0\leq\Phi_\beta''(y) \leq C_\beta e^{A(y)}A(y)^q.\tag{13}\] Indeed, \[A'(y)=2\beta yS(y)^{-\beta-1}, \qquad A''(y)=2\beta S(y)^{-\beta-1} +4\beta(\beta+1)y^2S(y)^{-\beta-2},\] and \(\Phi_\beta''=e^A((A')^2+A'')\). Since \(A\geq1\), \(S^{-1}=A^{1/\beta}\), and \(|y|\leq1\), the displayed formulas give \((A')^2+A''\leq C_\beta'(A^{2+2/\beta}+A^{1+2/\beta}) \leq C_\beta'' A^q\). Next, for an integer \(d\geq2\), define the boundary layer \[B_d:=\big\{y\in(-1,1):1-|y|\leq6d^{-1/2}\big\}.\] On the complement of \(B_d\) in \((-1,1)\), define the scale-sensitive curvature envelope \[\label{eq:bump-Gd-main} G_d(y):=1+\sup_{|u-y|\leq d^{-1/2}}\Phi_\beta''(u),\tag{14}\] whereas on \(B_d\) set \(G_d(y)=0\). The boundary layer will be treated as a separate bad event.
Lemma 5 (Boundary estimate). There are constants \(C_\beta,c_\beta>0\) such that \[\label{eq:bump-boundary-tail} \mathbb{P}\{Y\in B_d\} \leq C_\beta\exp\{-\exp(c_\beta d^{\beta/2})\}.\tag{15}\]
Proof. If \(1-|y|\leq6d^{-1/2}\), then \(S(y)\leq12d^{-1/2}\), and hence \[\Phi_\beta(y)\geq\exp\big(\big(12d^{-1/2}\big)^{-\beta}\big) =\exp(c_\beta d^{\beta/2}).\] Integrating 11 over the boundary layer and absorbing its length and \(Z_\beta^{-1}\) into the constant proves the claim. ◻
Lemma 6 (Scale-sensitive curvature tail). There are constants \(C_\beta,c_\beta>0\) such that, for every \(d\geq2\), \[\label{eq:bump-G-tail} \mathbb{P}\{G_d(Y)>x\} \leq C_\beta\exp\left[-\frac{c_\beta x}{\ell(x)}\right] \qquad(1\leq x\leq d).\tag{16}\] Moreover, \[\label{eq:bump-G-second} \sup_{d\geq2}\mathbb{E}[ G_d(Y)^2]<\infty.\tag{17}\]
Proof. We may assume that \(d\) and \(x\) exceed constants depending on \(\beta\); all remaining cases are absorbed by changing \(C_\beta\).
Fix \(y\in (-1,1)\setminus B_d\) and \(|u-y|\leq d^{-1/2}\). By the Mean Value Theorem, \(|A(u)-A(y)|=|A'(\xi)|\,|u-y|\) for some \(\xi\) between \(u\) and \(y\). We have \(S(\xi)\geq S(y)-|S(\xi)-S(y)| \geq S(y)-2d^{-1/2}\geq\frac{2}{3}S(y)\), because \(|\xi-y|\leq d^{-1/2}\), \(|\xi+y|\leq2\), and \(S(y)\geq1-|y|>6d^{-1/2}\). Therefore \[\label{eq:bump-A-comparison} A(u)\leq A(y)+C_\beta d^{-1/2}\,S(y)^{-\beta-1} =A(y)+C_\beta d^{-1/2}\,A(y)^{1+1/\beta}.\tag{18}\] Suppose that \[\label{eq:bump-T-small} e^{A(y)}\leq a_\beta\frac{x}{\ell(x)},\tag{19}\] where \(a_\beta>0\) will be chosen small. Since \(x\leq d\), this implies \(A(y)\leq \log x\leq\log d\). Consequently, \[d^{-1/2}\,A(y)^{1+1/\beta} \leq d^{-1/2}(\log d)^{1+1/\beta}.\] It follows from 18 , after enlarging the lower constant on \(d\), that \(e^{A(u)}\leq2e^{A(y)}\) and \(A(u)\leq2A(y)\). In view of 13 and 19 , \[G_d(y)\leq C_\beta' e^{A(y)}A(y)^q \leq C_\beta'' a_\beta x.\] Thus, choosing \(a_\beta\) sufficiently small showed that \[\label{eq:the32last32implication} G_d(y)>x\quad implies \quad e^{A(y)}>a_\beta\frac{x}{\ell(x)}.\tag{20}\] Further, the definition of the density gives \[\label{eq:bump-T-tail-main} \mathbb{P}\{e^{A(Y)}>s\}\leq C_\beta e^{-s}.\tag{21}\] Indeed, \[\mathbb{P}\{e^{A(Y)}>s\} =\frac{1}{Z_\beta}\int_{\{y:\Phi_\beta(y)>s\}} e^{-\Phi_\beta(y)}\,dy \leq\frac{2}{Z_\beta}e^{-s}.\] Therefore, 20 and 21 prove 16 .
For the second moment, if \(y\in (-1,1)\setminus B_d\) and \(|u-y|\leq d^{-1/2}\), then \(S(u)\geq S(y)/2\), whence \(A(u)\leq2^\beta A(y)\). By 13 , \[G_d(y)^2\leq C_\beta\, e^{2^{\beta+1}A(y)}A(y)^{2q}.\] Writing \(T(y)=e^{A(y)}\), the non-constant term on the right is a fixed polynomial in \(T(y)\) and \(\log T(y)\). Its expectation under the density proportional to \(e^{-T(y)}\) is bounded by a constant depending only on \(\beta\). This proves 17 . ◻
Lemma 7 (Almost-exponential average tail). Let \(Z_1,\ldots,Z_d\) be independent copies of \(G_d(Y)\). There are \(K_\beta,C_\beta,c_\beta>0\) such that \[\label{eq:bump-average-tail} \mathbb{P}\left\{\frac{1}{d}\sum_{j=1}^dZ_j>K_\beta\right\} \leq C_\beta \exp\left[-\frac{c_\beta d}{\ell(d)}\right].\tag{22}\]
Proof. By Lemma 6 and a union bound, \[\label{eq:bump-max-tail} \mathbb{P}\left\{\max_{j\leq d}Z_j>d\right\} \leq C_\beta d\exp[-c_\beta d/\ell(d)] \leq C_\beta'\exp[-c'_\beta d/\ell(d)].\tag{23}\] Put \(\lambda=c_0/\ell(d)\), where \(c_0>0\) is sufficiently small. The tail-integration identity and 16 yield \[\begin{align} \mathbb{E}e^{\lambda\min\{Z_j,d\}} &=1+\lambda\int_0^d e^{\lambda x}\mathbb{P}\{Z_j>x\}\,\mathrm{d}x\\ &\leq1+C_\beta\lambda \left(1+\int_1^\infty \exp\left[-\frac{c_\beta x}{2\ell(x)}\right]\,\mathrm{d}x\right)\\ &\leq\exp(C'_\beta\lambda). \end{align}\] Here we used \(\ell(x)\leq\ell(d)\) for \(1\leq x\leq d\), so that \(\lambda x\leq c_0x/\ell(x)\). Chernoff’s bound gives \[\mathbb{P}\left\{\sum_{j=1}^d\min\{Z_j,d\}>K_\beta d\right\} \leq\exp[-(K_\beta-C'_\beta)\lambda d].\] Choose \(K_\beta>2C'_\beta\) and combine this with 23 . ◻
Proposition 6 (Weighted scale-sensitive curvature tail). There are constants \(K'_\beta,C_\beta,c_\beta>0\) such that, for every \(M\geq1\), every \(w_1,\ldots,w_M\) satisfying 8 , and independent copies \(Y_1,\ldots,Y_M\) of \(Y\), \[\label{eq:weighted-bump-tail} \mathbb{P}\left\{\sum_{i=1}^M w_iG_d(Y_i)>K'_\beta\right\} \leq C_\beta \exp\left[-\frac{c_\beta d}{\ell(d)}\right].\qquad{(3)}\]
Proof. Apply Lemma 4 with \(Z=G_d(Y)\) and use Lemma 7. The second-moment factor is bounded uniformly in \(d\) by 17 . The square root in 9 only changes \(c_\beta\), and the additive one only changes the threshold. ◻
Proposition 7 (Bump-law balance estimate). For \(L=L_\beta\) sufficiently large, the following holds for every \(m\geq1\). Let \(Y_1,\ldots,Y_m\) be independent with \(Y_i\sim h_\beta\) and put \(X=(LY_1,\ldots,LY_m)\). If \(v\in\mathbb{R}^n\) satisfies \(\left\lVert v\right\rVert_2\leq1\) and \(\left\lVert v\right\rVert_\infty\leq d^{-1/2}\) then \[\label{eq:bump-balance-prob} \mathbb{P}\{X\text{ is not v-balanced}\} \leq C_\beta \exp\left[-\frac{c_\beta d}{\log^q(ed)}\right].\qquad{(4)}\]
Proof. In this proof, we use subscript notation \(v_i\) for components of \(v\). Put \(z_i=v_i/L\) and \(w_i=v_i^2\) for \(1\leq i\leq m\). Then \(|z_i|\leq d^{-1/2}\), \(0\leq w_i\leq d^{-1}\), and \(\sum_{i=1}^m w_i\leq1\). Next, discard the event that \(Y_i\in B_d\) for some \(i\in\operatorname{supp}v\). There are at most \(d\) such coordinates, so Lemma 5 bounds its probability by \[C_\beta d\exp\{-\exp(c_\beta d^{\beta/2})\} \leq C_\beta\exp[-c_\beta d/\log^q(ed)].\] On its complement all \(Y_i\pm z_i\) lie in \((-1,1)\). The central-difference formula and 14 give \[\begin{align} D_v(X) &=\sum_{i=1}^m\bigl(\Phi_\beta(Y_i+z_i)+\Phi_\beta(Y_i-z_i) -2\Phi_\beta(Y_i)\bigr) \leq\sum_{i=1}^m z_i^2G_d(Y_i) =\frac{1}{L^2}\sum_{i=1}^m w_iG_d(Y_i). \end{align}\] Let \(K'_\beta\) be the threshold in Proposition 6 and choose \(L^2\geq K'_\beta/(2\log2)\). Except on the event in ?? , we then have \(D_v(X)\leq2\log2\). Lemma 2 implies that \(X\) is \(v\)-balanced. Combining the two exceptional-event estimates proves the proposition. ◻
Proof of Theorem 1. Normalize \(v_t=a_t/\sqrt d\). Since each \(a_t\) is \(d\)-sparse with entries in \([-1,1]\), our goal under this normalization is to prove a discrepancy bound of order one. Apply Proposition 4 with the product law from Proposition 7. Its support radius depends only on \(\beta\), so the normalized prefix discrepancy is at most \(6L_\beta\) outside an event of probability \[C_\beta T\exp[-c_\beta d/\log^q(ed)].\] Scaling back by \(\sqrt d\), choosing \(\beta\geq2/\eta\), and adjusting the constants gives 2 . ◻