1. Place potatoes in a large pot with enough water to cover by several inches. Bring to a boil over high heat, add 2 tablespoons salt, and reduce to a gentle boil. Cook until potatoes are tender when pierced with a knife, about 15 minutes. Drain into a colander. Using gloves or paper towels to protect your hands, cut the potatoes into 3/4-inch-thick wedges while still hot. Drizzle with 1/4 cup vinegar, and set aside.

2. Place radishes and red onion in a large bowl; add remaining 1/4 cup vinegar. Let stand 30 minutes, stirring every 5 minutes.

3. Place cucumbers in a colander over a bowl; sprinkle with 1/2 teaspoon salt. Set aside.

4. When potatoes are cool, add the red-onion mixture. Add reserved cucumbers, dill, olive oil, remaining 1/2 teaspoon salt, and pepper, and toss to combine. Chill until ready to serve. Garnish with arugula.