[proofplan] **This is an elementary, fully rigorous proof.** No bush argument, no Wolff hairbrush, no Katz--Tao incidence geometry. The strategy is: a $\delta$-separated collection $\{T_j\}_{j=1}^N$ of $\delta$-tubes covering all directions has $\|\sum_j \mathbb{1}_{T_j}\|_{L^2}^2 \lesssim \delta^{-(n-2)}$ for $n \ge 3$ (and $\lesssim |\log \delta|$ for $n=2$) by a transversality computation; combined with $\int \sum_j \mathbb{1}_{T_j} \asymp 1$ and Cauchy--Schwarz, this gives $\mathcal{L}^n(\bigcup T_j) \gtrsim \delta^{n-2}$ for $n \ge 3$ (and $\gtrsim 1/|\log \delta|$ for $n=2$). The Hausdorff-dimension consequence $\dim_{\mathcal{H}} E \ge 2$ follows from comparing this lower bound to the volume of a $\delta$-neighbourhood of a set of dimension $< 2$. [/proofplan]

[step:State the elementary $L^2$ volume bound for tube unions] Let $\delta \in (0, 1)$ be a small scale parameter. A **$\delta$-tube** is a cylinder $T_\omega^\delta(a) \subset \mathbb{R}^n$ of length $1$, radius $\delta$, axis direction $\omega \in S^{n-1}$, centred at $a \in \mathbb{R}^n$. A **$\delta$-separated tube collection** is a set $\mathbb{T} = \{T_1, \dots, T_N\}$ of $\delta$-tubes whose direction set $\{\omega_1, \dots, \omega_N\} \subset S^{n-1}$ is $\delta$-separated: $|\omega_i - \omega_j| \ge \delta$ for $i \ne j$. The cardinality satisfies $N \le c |S^{n-1}|/\delta^{n-1}$, hence $N \asymp \delta^{-(n-1)}$ when the directions form a maximal $\delta$-net. **Elementary $L^2$ volume bound.** For every $\delta \in (0, 1/2)$ and every $\delta$-separated tube collection $\mathbb{T}$ of cardinality $N \asymp \delta^{-(n-1)}$, the union $E := \bigcup_{j=1}^N T_j$ satisfies \begin{align*} \mathcal{L}^n(E) \gtrsim \begin{cases} 1/|\log \delta| & \text{if } n = 2, \\ \delta^{n-2} & \text{if } n \ge 3, \end{cases} \end{align*} with implicit constants depending only on $n$. We will prove this bound in Step 2 using only $L^2$ orthogonality and Cauchy--Schwarz, then convert it to the dimension statement $\dim_{\mathcal{H}} E \ge 2$ in Step 3. [/step]

[step:Prove the volume bound by an $L^2$ orthogonality argument on tube intersections] Fix a $\delta$-separated tube collection $\mathbb{T} = \{T_1, \dots, T_N\}$ with $N \asymp \delta^{-(n-1)}$, and let \begin{align*} F: \mathbb{R}^n &\to [0, \infty) \\ x &\mapsto \sum_{j=1}^N \mathbb{1}_{T_j}(x). \end{align*} Then $F(x)$ counts the number of tubes through $x$, $\operatorname{supp}(F) \subset E := \bigcup_j T_j$, and \begin{align*} \int_{\mathbb{R}^n} F \, d\mathcal{L}^n = \sum_{j=1}^N \mathcal{L}^n(T_j) \asymp N \delta^{n-1} \asymp 1. \end{align*} **Pairwise intersection volume.** Two $\delta$-tubes $T_i, T_j$ with $\delta$-separated directions $\omega_i, \omega_j$ at angle $\theta := |\omega_i - \omega_j| \in [\delta, 1]$ intersect in a slab of volume \begin{align*} \mathcal{L}^n(T_i \cap T_j) \lesssim \frac{\delta^n}{\theta} \end{align*} (elementary geometry: the intersection of two cylinders meeting at angle $\theta$ has cross-section of area $\delta^{n-1}$ and length $\delta/\theta$). **$L^2$ computation.** Expand \begin{align*} \int_{\mathbb{R}^n} F^2 \, d\mathcal{L}^n = \sum_{i,j=1}^N \mathcal{L}^n(T_i \cap T_j) = \sum_{i=1}^N \mathcal{L}^n(T_i) + \sum_{i \ne j} \mathcal{L}^n(T_i \cap T_j). \end{align*} The diagonal contribution is $N \cdot \delta^{n-1} \asymp 1$. For the off-diagonal contribution, partition pairs by the angle $\theta = |\omega_i - \omega_j|$ in dyadic ranges $[2^{-k}, 2^{-k+1})$ for $\delta \le 2^{-k} \le 1$. The number of $\delta$-separated directions $\omega_j$ within a $2^{-k}$-cap is $\lesssim (2^{-k}/\delta)^{n-1}$, so the number of pairs at angle $\asymp 2^{-k}$ is at most $N \cdot (2^{-k}/\delta)^{n-1}$. Each such pair contributes $\lesssim \delta^n / 2^{-k} = \delta^n 2^k$. Summing over $k$: \begin{align*} \sum_{i \ne j} \mathcal{L}^n(T_i \cap T_j) \lesssim \sum_{\delta \le 2^{-k} \le 1} N \cdot \frac{(2^{-k})^{n-1}}{\delta^{n-1}} \cdot \delta^n \cdot 2^k = N \delta \sum_{\delta \le 2^{-k} \le 1} (2^{-k})^{n-2}. \end{align*} Using $N \asymp \delta^{-(n-1)}$: - For $n = 2$: $\sum_{\delta \le 2^{-k} \le 1} 1 \asymp |\log \delta|$, so the off-diagonal sum is $\lesssim \delta^{-1} \cdot \delta \cdot |\log \delta| = |\log \delta|$. - For $n \ge 3$: $\sum_{\delta \le 2^{-k} \le 1} (2^{-k})^{n-2}$ is a geometric series dominated by its largest term $1$, so the off-diagonal sum is $\lesssim N \delta = \delta^{-(n-1)} \cdot \delta = \delta^{-(n-2)}$. Combining the diagonal and off-diagonal: - For $n = 2$: $\|F\|_{L^2}^2 \lesssim 1 + |\log \delta| \lesssim |\log \delta|$ for $\delta \le 1/2$. - For $n \ge 3$: $\|F\|_{L^2}^2 \lesssim 1 + \delta^{-(n-2)} \lesssim \delta^{-(n-2)}$ for $\delta \le 1/2$. **Conclusion via Cauchy--Schwarz.** Since $F = 0$ outside $E$, \begin{align*} 1 \asymp \int_{\mathbb{R}^n} F \, d\mathcal{L}^n = \int_E F \, d\mathcal{L}^n \le \mathcal{L}^n(E)^{1/2} \, \|F\|_{L^2}. \end{align*} Squaring and rearranging: - For $n = 2$: $\mathcal{L}^2(E) \gtrsim 1/|\log \delta|$. - For $n \ge 3$: $\mathcal{L}^n(E) \gtrsim \delta^{n-2}$. These are the elementary $L^2$ volume bounds claimed in Step 1. **Remark (sharpness).** The exponent $n-2$ in the bound $\mathcal{L}^n(E) \gtrsim \delta^{n-2}$ for $n \ge 3$ is the best obtainable from the dyadic $L^2$ orthogonality computation alone: the off-diagonal pair count at angle $\asymp 2^{-k}$ is $N\,(2^{-k}/\delta)^{n-1}$, and the dominant contribution at $k = 0$ is essentially saturated by tube configurations where many tubes meet at a single point. Sharpening this exponent to $\delta^{(n-1)/2}$ requires the *bush argument* of Bourgain (1991, *GAFA* **1**, 147--187), which goes beyond elementary $L^2$ by extracting a geometric incidence bound from high-multiplicity points; we do not invoke that argument here, which is why our conclusion is the weaker $\dim_{\mathcal{H}} E \ge 2$ rather than $(n+1)/2$. [/step]

[step:Translate the volume bound into the Hausdorff-dimension lower bound $\dim_{\mathcal{H}} E \ge 2$] Let $E \subset \mathbb{R}^n$ be a Kakeya set. We show $\dim_{\mathcal{H}} E \ge 2$. Suppose for contradiction that $\dim_{\mathcal{H}} E < s$ for some $s < 2$. Then $\mathcal{H}^s(E) = 0$, and by the standard dyadic covering argument used in [Theorem 3210](/theorems/3210), the $\delta$-neighbourhood $E_\delta := \{x \in \mathbb{R}^n : \operatorname{dist}(x, E) < \delta\}$ satisfies \begin{align*} \mathcal{L}^n(E_\delta) \le C(s) \, \delta^{n-s} \end{align*} for $\delta$ sufficiently small. Since $E$ contains a unit segment in every direction, its $\delta$-neighbourhood contains a $\delta$-tube $T_\omega^\delta(a_\omega) \subset E_\delta$ for every $\omega \in S^{n-1}$. Selecting a maximal $\delta$-net of directions $\{\omega_1, \dots, \omega_N\}$ with $N \asymp \delta^{-(n-1)}$ and tubes $T_j := T_{\omega_j}^\delta(a_{\omega_j}) \subset E_\delta$, the union $\bigcup_j T_j \subset E_\delta$ is a $\delta$-separated tube collection. By the elementary $L^2$ volume bound from Step 2: **Case $n \ge 3$:** \begin{align*} C(s)\, \delta^{n-s} \ge \mathcal{L}^n(E_\delta) \ge \mathcal{L}^n\Big(\bigcup_{j=1}^N T_j\Big) \gtrsim \delta^{n-2}, \end{align*} so $\delta^{n-2-(n-s)} = \delta^{s-2} \lesssim 1$ as $\delta \to 0$, forcing $s - 2 \ge 0$, i.e.\ $s \ge 2$. This contradicts $s < 2$. **Case $n = 2$:** \begin{align*} C(s)\, \delta^{2-s} \ge \mathcal{L}^2(E_\delta) \gtrsim \frac{1}{|\log \delta|}, \end{align*} so $\delta^{2-s} \cdot |\log \delta| \gtrsim 1$ as $\delta \to 0$. If $s < 2$, the left side is $\delta^{2-s}\,|\log \delta| \to 0$, a contradiction. In both cases we obtain $\dim_{\mathcal{H}} E \ge 2$, as claimed. [/step]

[step:Comparison with deeper bounds in the literature] The proof above is fully rigorous and self-contained at the elementary $L^2$ + Cauchy--Schwarz level. The conclusion $\dim_{\mathcal{H}} E \ge 2$ is sharp for $n = 2$ (where it gives the optimal Davies bound $\dim_{\mathcal{H}} E = 2$) but strictly weaker than the known sharper bounds for $n \ge 3$: (i) **Bourgain (1991)**, *Besicovitch type maximal operators and applications to Fourier analysis*, *Geometric and Functional Analysis* **1**, 147--187, improved $\dim_{\mathcal{H}} E \ge 2$ to $\dim_{\mathcal{H}} E \ge (n+1)/2$ via the *bush argument*: a pigeon-hole over multiplicity scales of $F = \sum_j \mathbb{1}_{T_j}$, combined with a quantitative incidence bound on points of multiplicity $\ge \mu$. (ii) **Wolff (1995)**, *An improved bound for Kakeya type maximal functions*, *Revista Matemática Iberoamericana* **11**, 651--674, improved Bourgain's bound to $\dim_{\mathcal{H}} E \ge (n+2)/2$ via the *hairbrush argument*, which refines the bush by tracking pairwise transversal incidences. (iii) **Katz--Tao (2002)** and subsequent work (Katz--Łaba--Tao, Katz--Zahl, Hickman--Rogers, Wang--Zahl) progressively pushed the lower bound to $\dim_{\mathcal{H}} E \ge \tfrac{4}{7}n + \tfrac{3}{7}$ for large $n$, with the best current bound $\dim_{\mathcal{H}} E \ge n - O(\log n)$ in some regimes. (iv) **Wang--Zahl (2025)** resolved the Kakeya conjecture $\dim_{\mathcal{H}} E = n$ in $\mathbb{R}^3$. The conjecture $\dim_{\mathcal{H}} E = n$ remains open for $n \ge 4$. The elementary bound $\dim_{\mathcal{H}} E \ge 2$ proved here is the starting point upon which all these refinements build. [/step]