O2 Sensor Problems of Hyundai Veloster

Car often fails to start when turning the ignition key. The lights, and radio turn on when the ignition key is in, but when I turn the key to start I hear a click and the car is dead. If I repeat this several times the car may start normally. The problem stated several weeks ago. I have gone to the dealer twice. The first time they claimed it was the battery. They changed the battery (it cost me $100). This solved the problem for 2-3 days (or perhaps it was the warmer weather, see below). A week later the problem came back and the engine light came on. They replaced the oxygen sensor. This fixed the engine light issue. The car was ok for a day. The problem with failing start is back and it is getting worst. Yesterday it took me 5 minutes in the morning and 25 minutes at night to start the car. I got the feeling that the problem worsens in cold weather (I told that to the dealer the first time I brought it in, but they just ignored me). Each time I took it to the dealer the whether happened to be warmer (above 0 celsius). Last couple of days the temperature was below -3 celsius. Yesterday I parked the car inside my garage. This morning, with a temperature +4 celsius in my garage, the car started at the first try. I waited 10 minutes and tried again and the car again started at the first try. I will go to the dealer again this weekend. I tried different ways of starting the car with no success (1-clutch engaged and first, 2-clutch engaged and neutral, 3-neutral and no clutch, 3-different variations of radion on/off, as well as heater fan on, or off. The following may have been there always but now I notice since I am more aware of car sounds. There is a beeping sound when I enter the key and turn the dashboard on. That beeping sound last for about 10 seconds and ends with a click. The same happens after the car engine is turn off.

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